.Division 
Range 
Shelf 
Received 


1S7// 


No.    12. 


PROFESSIONAL   PAPERS 


OF 


THE  CORPS  OF  ENGINEERS 


OF 


THE  UNITED  STATES  ARMY. 


PUBLISHED 


BY    AUTHORITY    OF    THE    SECRETARY    OF   WAR. 


HEADQUARTERS    CORPS   OF    ENGINEERS. 


1873. 


TABLES   AND    FORMULAE 


SURVEYING,    GEODESY, 


PRACTICAL  ASTRONOMY, 


ELEMENTS  FOR  THE  PROJECTION  OF  MAPS, 


INSTRUCTIONS  FOR  FIELD  MAGNETIC  OBSERVATIONS. 


v-t  r 
;  ^.Cv^cv 


Third  Edition,  Revised  and  Enlarged. 


WASHINGTON: 
GOVERNMENT    PRINTING     OFFICE. 

1873. 


OFFICE  OF  THE  CHIEF  OF  ENGINEERS, 

Washington,  February  24,  1873. 

GENERAL  :  In  preparing  this  third  edition  of  a  volume  com 
piled  in  1849  for  tne  use  of  the  Corps  of  Topographical  Engineers, 
when  an  officer  in  that  corps,  I  have  made  such  additions  and 
corrections  as  experience  has  suggested  and  the  requirements  of 
the  service  seemed  to  demand. 

Intended  more  especially  for  field-use  by  officers  engaged  in 
surveys  or  explorations,  and  aspiring  to  no  further  merit  than  that 
of  accuracy,  utility,  and  convenience,  it  is  submitted  with  the 
hope  that  it  may  continue  to  be  favorably  received  by  those  who 
may  have  occasion  to  refer  to  it. 

The  revision  was  undertaken  at  the  suggestion  of  others,  and 
not  without  reluctance.     I  beg  indulgence  for  its  defects. 
Very  respectfully, 

THOS.  J.  LEE. 
Brigadier-General  A.  A.  HUMPHREYS, 

Chief  of  Engineers,  United  States  Army. 


In  the  additions  and  corrections  introduced  into  this  volume,  besides  the 
authorities  named  in  the  text,  I  have  availed  myself,  in  the  article  on  the 
Ganging  of  Rivers,  of  notes  by  MAJOR  ABBOT,  Corps  of  Engineers,  on  the 
practical  gauging  of  rivers,  printed  in  the  proceedings  of  the  Essay ons  Club 
at  Willefs  Point. 

The  article  on  Trigonometrical  Leveling  is  taken  principally  from  Appendix 
No.  7  of  the  Coast-Surjey  report  for  1868,  by  ASSISTANT  R.  D.  CUTTS,  United 
States  Coast  Survey. 

I  have  also  made  use  of  Appendices  Nos.  9,  io,'  and  II  of  Coast-Survey 
report  of  1866,  07  ASSISTANT  C.  A.  SCHOTT,  United  States  Coast  Survey,  in 
the  revision  of  the  portions  relating  to  the  use  of  the  transit-instrument,  of  the 
zenith-telescope,  and  the  determination  of  astronomical  azimuths. 

The  article  on  Longitude  by  Lunar  Culminations  was  prepared  in  1858 /I?/ 
the  Topographical  Bureau,  by  PROFESSOR  BARTLETT,  United  States  Military 
Academy. 

I  am  indebted  to  CAPTAIN  ERNST  and  LIEUTENANT  MERCUR,  Corps  of  En 
gineers,  and  to  ASSISTANTS  WOODWARD  and  WRIGHT,  Survey  of  the  Lakes, 
for  suggestions  and  corrections  ;  &  LIEUTENANT  MERCUR  for  the  article  on 
Probable  Errors,  &c.;  and  to  CAPTAIN  RAYMOND,  Corps  of  Engineers,  for 
the  valuable  contribution  on  Magnetic  Field  Observations,  -which  forms  the 
Appendix;  and  I  have  to  regret  that  LiEUTENANT-CoLONEL  WILLIAMSON  was 
prevented  by  absence  from  the  country  from  preparing  a  suitable  set  of 
hvpsometrical  tables,  which  he  had  consented  to  do. 

T.  J.  L, 


CONTENTS. 


PART  I.— MISCELLANEOUS. 

Page. 

Trigonometry,  equivalent  expressions 3 

Solution  of  plane  triangles 5 

Solution  of  spherical  triangles 6 

Multiple  arcs -. 7 

Trigonometrical  lines 8 

Differentials  of  trigonometrical  lines 8 

Arcs  in  parts  of  radius 9 

Signs  of  trigonometrical  lines ..  9 

Weights  and  measures  of  the  United  States 10 

Miscellaneous  measures 12 

Weights  and  volumes  of  different  substances 13 

Component  parts  of  the  Army  ration 15 

Metric  system  of  weights  and  measures 16 

Metrical  equivalents 17 

Foreign  measures  of  length 18 

Foreign  itinerary  measures 19 

Table  for  converting  metres  into  toises  and  French  and  English  feet..  20 

Table  for  converting  English  feet  into  toises,  metres,  and  French  feet.  21 

Analytical  expressions  for  lines,  surfaces,  and  solids 22 

Progression , 26 

Force  of  gravity , 26 

Land-surveying  with  compass  and  chain 27 

Traverse  table 29 

Table  for  converting  chains  into  feet,  and  vice  versa 44 

To  trace  railroad  curves , 46 

Gauging  of  rivers c^ 

Discharge  of  water  through  pipes 61 

Four-place  logarithms  of  numbers 62 

Four-place  logarithms  of  sines,  tangents,  &c 64 

Table  of  squares  and  square  roots  from  I  to  looo • 68 

Table  to  facilitate  interpolation  by  differences 79 

PART  II.— GEODESY. 

Reduction  to  center  of  station 81 

Correction  for  phase 81 

Spherical  excess g2 

Reduction  of  bases 83 

Correction  for  temperature  in  metallic  rods 85 


Vlll  CONTENTS. 


Page. 

Measurement  of  distances  by  sound 86 

Problem  of  the  three  points 87 

Formulae  for  computing  the  principal  geodetic  quantities  depending  on 

the  spheroidal  figure  of  the  earth 88 

Bessel's  magnitude  and  figure  of  the  earth 1 89 

Relative  length  of  the  yard  and  the  metre - 90 

Numerical  values  of  Bessel's  terrestrial  elements  in  English  yards 92 

Constant  logarithms  useful  in  geodetic  computations 92 

Formulae  for  computing  geodetic  latitudes,  longitudes,  and  azimuths. .  93 

Measurement  of  distances,  from  astronomical  determinations 95 

Clarke's  magnitude  and  figure  of  the  earth 97 

Forms  for  record  and  computation 98 

Logarithmic  values,  in  yards,  of  the  normal  in  different  latitudes 100 

Logarithmic  values,  in  yards,  of  the  radius  of  curvature  of  the  me 
ridian  in  different  latitudes 103 

Formulae  for  the  polyconic  projection  of  maps 106 

Co-ordinates,  in  yards,  for  the  projection  of  maps 108 

Values  of  arcs  of  parallel,  in  yards 127 

Values  of  meridional  arcs,  in  yards 127 

Lengths,  in  miles,  of  degrees  of  latitude  and  longitude 137 

Co-ordinates,  in  statute  miles,  for  the  projection  of  maps 138 

Trigonometrical  measurement  of  heights  .  — —  ....  14° 

Apparent  and  true  level — 144 

Table  of  corrections  for  curvature  and  refraction 144 

,  Table  for  reducing  inclined  measures  to  horizontal 145 

/  Table  for  the  ratio  of  slopes 145 

Barometrical  measurement  of  heights 146 

Thermometrical  measurement  of  heights 181 

Table  for  converting  Fahrenheit's  scale  of  the  thermometer  to  Reaumur's 

and  the  centesimal 182 

PART  III.— ASTRONOMY. 

Of  sidereal  and  solar  time 185 

-/   To  find  the  time  by  an  altitude  of  the  sun  or  of  a  star 187 

Horizontal  sun-dial 188 

Table  for  converting  sidereal  into  mean  solar  time 19° 

Table  for  converting  mean  solar  into  sidereal  time .., 191 

Table  for  converting  space  into  time 192 

Table  for  converting  time  into  space 196 

Table  for  converting  AR.  in  arc  into  mean  time 1 98 

Table  for  converting  mean  time  into  AR.  in  arc 201 

Form  for  record  and  computation  of  the  determination  of  the  time  by 

altitudes  of  stars 204 

Computation  of  an  example  of  this  method 206 

To  find  the  time  by  equal  altitudes  of  the  sun 207 

Table  for  determining  the  equation  of  equal  altitudes 208 

Computation  of  an  example  of  this  method 217 


CONTENTS.  IX 


Page. 

Form  for  record  and  computation  of  this  method 218 

Table  of  the  sun's  parallax  in  altitude 220 

Table  of  decimals  of  an  hour 220 

The  transit-instrument,  correction  to  observed  transits 221 

Form  for  record  and  computation  of  observed  transits 224 

Example  of  the  method  of  computing  transit  corrections 225 

Table  to  facilitate  the  reduction  of  transit  observations 226 

Reduction  of  transits  by  least  squares 228 

Refraction 230 

Table  of  mean  refraction 23 1 

Bessel's  refraction  table 235 

To  determine  the  latitude  by  meridional  altitudes 237 

To  determine  the  latitude  by  circum-meridian  altitudes 238 

Tables  for  reduction  to  the  meridian 240 

Form  for  record  and  computation  of  the  determination  of  the  latitude 

by  circum-meridian  altitudes 248 

Computation  of  some  of  the  quantities  in  the  preceding  method 250 

To  determine  the  latitude  by  altitudes  of  circumpolar  stars 251 

Form  for  record  and  computation  of  the  method  of  determining  the 

latitude  by  altitudes  of  Polaris 252 

Computation  of  an  example  of  this  method 254 

To  determine  the  latitude  with  the  transit-instrument  by  transits  of 

stars  over  the  prime  vertical 255 

To  determine  the  latitude- with  the  zenith  and  equal-altitude  telescope.  258 

Form  for  record  and  computation  of  this  method 264 

To  find  the  azimuth  of  the  sun  or  of  a  star 265 

To  find  the  amplitude  of  the  sun  or  of  a  star 266 

To  determine  the  true  meridian  by  equal  altitudes  of  the  sun 266 

To  find  the  azimuth  of  Polaris  at  its  greatest  elongation 267 

Corrections  to  observed  azimuths 268 

Correction  for  run  in  reading  microscopes 270 

To  determine  the  longitude  by  lunar  culminations 271 

To  determine  the  longitude  by  the  electric  telegraph 280 

Formulae  for  probable  error  and  precision 288 

Peirce's  criterion  for  the  rejection  of  doubtful  observations 289 

Geographical  Positions 292 

APPENDIX. 

Field  magnetic  observations , „ 295 


ERRATA 

To  the  3d  edition  of  Tables  and  Formula,  Professional  Papers  No.  12. 


Page     10.  For  See.  XXXIV,  read  Sec.  XXX VII. 

Page     1.4.  Limestone,  for  197.5,  read  179.5,  and  for  11.355,  read  12.462. 

Page     47.  In  Chord-Deflection,  for  is  "therefore  double",  read  "may  be 

assumed  as  double"  the  tangent  deflection. 
Page     85.  Hassler's  expansion    of  iron    bar,  for  0.000240687260,  read 

0.000250687260. 

Page     96.  In  value  u"  Cos.  Z,  strike  out  Cos2!/  from  the  numerator. 
Page     99.  For  log.  rf  M  ±=2.9060529,  read  log.  rf  M=2.9060531. 
rfZ=2.7619618,       •    log.  JZ=2.761 9620. 

Page  134.  In  4th  column,  5th  line  of  table,  for  449.4,  read  499.4. 
Page  135.  In  7th  column,  4th  line  of  table,  for  331.0,  read  231.0. 
Page  142.  For  0.000000667,  read  0.0000000667. 
Page  159.  Omit  minus  sign  at  foot  of  last  column. 
Page   I  OS.  Top  line  of  last  column  for  +0.0958,  read  +0.0956. 
Page   168.   10th  line  of  table,  for  +0.9967,  read +0.0967. 
Page  170.  At  head  of  table,  read  1).  ^60384.3  log.  — X  &c. 

Page  229.  In  last  line  and  last  figure,  for  .06,  read  .09. 

Page  250.  In  lines  13,  15  and  18,  increase  the  values  by  one  tenth  of  a 

second. 

Page  268.  In  value  of  r  in  example,  for  2m.  30.3s,  read  2m.  20.3*. 

Page  288.  For  E,  read  e. 

Page  288.  Forjp,  p/  etc.,  read  p',  p"  etc. 

Page  289.  For  /ce>y,  read  «£<u. 

Page  290.  For  1  and  10  in  first  example,  read  1  and  11. 


ERRATA. 


Page  10.     For  "See  XXXIV"  read  "See  XXXVII." 

Page  85.     Hassler's  expansion  of  iron    bar,  for  "0.000240687260"  read 
'0.000250687260." 
Page  288.     For  E  read  e. 
Page  288.     For/,  /,  &c.,  read/,  /",  &c. 


TABLES  AND   FORMULAE 


PART    I. 


MISCELLANEOUS. 


TRIGONOMETRY. 


I. — Equivalent  Expressions. 

sin2  x  -f  cos2  x  =  i 

sin  x  =  cos  x  tan  x 

__cos# 
~~ cotx 

=  V  I  —  COS2  X 


V  i  4-  cot2  x 
=  2  sin  J  #  cos  J 
_         tan  # 

V  i  +  tan2  # 
i 

cosec  x 

~  tan  x 

=  sin  #  cot  x 


=::       i  —  sin2  x 
=  i  —  2  sin2  J  # 
=  cos2     x  —  sin2     x 


tan  #  = 


sec  x 
sin  ic 

COS  iC 

i 

cot  x 
sin  a 


V  i  —  sin2  ,7 

i  —cos  2  # 
sin  2  x 

sin  2  # 
i      cos  2  a? 


TRIGONOMETRY. 


I . — Equivalent  Expressions — C ontin u ed . 


cot  x  = 


tan  x 


sec  x  =  —  - 
cos  x 


cosec  x  =  — 

sin  .r 


versin  x  =  i  —  cos  x 
=  2  sin2  J  x 

co-versin  x  =  i  —  sin  x 
chord  x  =  2  sin  J  x 

sin  (A  i  B)  =  sin  A  cos  B  i  sin  B  cos  A 
cos  (A  =L  B)  =  cos  A  cos  B  ^  sin  A  sin  B 
sin  2  A  =  2  sin  A  cos  A 
cos  2  A  =  2  cos2  A  —  i 
=  1—2  sin2  A 
=  cos2  A  —  sin2  A 
2  cos2  J  A  =  i  +  cos  A 
2  sin2  J  A  =  i  —  cos  A 

tan  A  ^  tan  B 


tan(Ad.B)= 


tan  A  tan  B 


1 1  —  cos  A 


-j-  cos  A 

_  i  —  cos  A 
sin  A 

sin  A  ±  sin  B  =  2  sin  J  ( A  i  B)  cos  J  (A  =p  B) 
cos  A  +  cos  B  =  2  cos  J  (A  +  B)  cos  J  (A  —  B) 
cos  A  —  cos  B  =  2  sin  £  (A  -f  B)  sin  J  (B  —  A) 
sin2  A  —  sin2  B  =  sin  (A  -f  B)  sin  (A  —  B) 
cos2  A  —  sin2B  =  cos  (A  -f  B)  cos  (A  —  B) 


TRIGONOMETRY. 


I. — Equivalent  Expressions — Continued. 


-n       sin  (A  4:  B) 

tan  A  dL  tan  B  = 1-, 

cos  A  cos  B 


sin  A  sin  B 

sin  A  +  sin  B  _  tan  \  (A  +  B) 
sin  A  —  sin  B  ~~  tan  f(A  —  B) 

"~   A=tan2(450dLJA) 


sin  A 

sin  A 


cos  A 


JA) 


1  1  .  —  Solution  of  Plane  Triangles. 

In  the  following  formulas,  A,  B,  C,  represent  the  angles,  and 
a,  ft,  c,  the  sides  opposite,  respectively. 

i.  Any  plane  triangle: 


sin  A  _  sin  B  _  sin  C 
a  b  c 

tan_JJA_+B)  __        cot  ^  C 
tan  I  (A  -^B)"  "~  tan  J  (A  -  B) 


cos  A  A 


(  s  (s  —  a]  } 

=  J  -±- i  I 

(       ^<r       ) 


.  2 


TRIGONOMETRY. 


II.—  Solution  of  Plane  Wangles—  Continued. 

2.  Right-angled  triangles  : 

Making  A  =  90°  in  the  preceding,  they  become 

a2  =  b~  +  ^ 

b  —  a  sin  B  =  #  cos  C, 
c  =  (i  sin  C  =  a  cos  B 


tan  B=- 
c 

tanC  =  - 


HI—  Solution  of  Spherical  Triangles. 

.a,  b,  c,  represent  the  arcs,  and  A,  B,  C,  the  angles  opposite, 
i.   Oblique  spherical  triangles: 

sin  A  _sinJB  _  sin_C 
~sm"rtr"~'smT  ~~  sin  c 

cos  b  sin  (c  +  y) 
^  cos  a  =  -    —  ^   - 

^  cot  9"  =  tan  b  cos  A 

cos  B  sin  (C  —  c) 
- 


cot  (f  =  tan  B  cos  a 

sin  (C  +  ?) 


cot  a  tan  b  = 


sin 


cot  c?  = 


cot  A 


v  cos  b 

Napier's  Analogies. 

cosJ(A-B) 
tan  J  (<z  +  /;)  =  tan  J-  ^  AB 


sinJ(A-B) 

tan  4  («  -  ^)  =  tan  J  ^  sJjTf  (A  +  B) 

cos  J(«  —  ^) 
tan  J  (A  +  B)  =  cot  J  C  ^J(^fT) 

sin  J(«  -^) 
tan  i  (A  -  B)  =  cot  J  C  ^y(^-+  ^  ) 


TRIGONOMETRY. 


III.  —  Solution  of  Spherical  Triangles  —  Continued. 

sin  S  sin  (A  —  S) 
sin2J"  =        sin  B  sin~CT- 


cos       a  =  sin  B  sin 


sin  (B-S)sin  (C-S) 
sin  B  sin  C~ 

2  sin  S  sin  (A  —  S) 

tan  *  a   ~  "sin  (B-S)  sin  (C  -  S) 

s-  - 

sin       A= 


sin  b  sin  c 

sin  s  sin  (s  —  a] 

cos   *  A  = = — 7—^ — 

sin  b  sin  c 

2  sin  (s  —  b)  sin  (.r  —  c} 

tan  J  A  =      sin  s  sin  ( j  —  0) 

In  which  S  and  s  represent  the  half-sum  of  the  three  an 
gles  diminished  by  90°  and  the  half-sum  of  the  three  sides, 
respectively. 

2.  Right-angled  spherical  triangles,  a  being  the  hypothenuse: 

cos  a  =  cos  b  cos  c  cot  B  =  cot  b  sin  c 

cos  a  =  cot  B  cot  C  cot  C  =  cot  c  sin  b 

cos  B  =  sin  C  cos  b  tan  b  =  tan  B  sin  c 

cos  C  =  sin  B  cos  c  tan  c  —  tan  C  sin  b 

tan  b  =  tan  a  cos  C  sin    b  =  sin  «  sin  B 

tan   c  =  tan  «  cos  B  sin    c  =  sin  <?  sin  C 

\\.-Multiple  Arcs. 

sin  2  a?  =  2  sin  #  cos  ,r 

sin  3  x  =  2  sin  x  .  cos  2  x  -{-  sin  # 

cos  2  x  =  2  cos  .r  .  cos  #  —  i 

cos  3  x  =  2  cos  .r .  cos  2  .r  —  cos  j; 

2  tan  x 


tan  2  .r  = 


i  —  tan2  ,r 


i  —  tan  x  .  tan  2  .r 


TRIGONOMETRY. 


V. — Trigonometrical  Series. 


A3  A  5  A  7 

sin  A  =  A--A+^A—  __      _+etc. 
2-3     2,3.4.5     2.3 7 

A  2  A  4  A  6 

cos  A  =  i  -  A+_±l_ i '     +  etc. 

2          2.3.4        2 6 


tan  A  =  A  +  ^+-^+  V—+  etc. 
3       3-5      32-5-7 

'     A  4-s*n3  A     -3  sm5  ^     3*5  -n~-  4-  etc 
2.3      "^2.4.5    "  "    2.4.6.7 

=  tan  A  —  i  tan3  A  +  J  tan5  A  —  l  tan7  A.  +  etc. 

-.9.  —A. 

log  sin  A  =  lc 


M  =  logarithmic  modulus 

=  0.43429  45 

log  M  =  9,63778  43IJ3 

Differentials  of  Trigonometrical  Lines. 
d  sin  #  =  +  d  x  cos  x 
d  cos  ^  =  —  d  x  sin  # 


cos^  x 

d  cot  x  =  —    .    ® ~ 
sm-5  x 

d  sin2  j?  =  4-  2  d  #  sin  #  cos 

=  —  2  d  j?  sin  ^  cos 

2  d  #  tan  x 
cos2  # 


d  tan2  x  =  + 


2  d  x  cot  ,o 

d  COt2  Xi=—  — 


sin2  x 


TRIGONOMETRY. 


VI. — Ratio  of  the  Circumference  of  a  Circle  to  its  Diameter. 

?r  =  3. 14159  26535  898 

log  71  =  0.49714  98726  941 

The  radius  being  unity,  the  number  of  degrees  in  an  arc  equal 

to  radius  =r°=^-  = — —< D  =  57°-29578  =  57°  i7/44//-8. 

TT         arc  i° 

The   number  of  minutes  =  r1  =-       -  = r,  or 


sin  i' 

,,_648ooo"_ 


The 


number  of  seconds  —  r"  ="H       21_  =  .-._* 

-  sin  i" 


log  r°  =  1.75812   26324  09172 
comp  log  r°  =8.24187   73675  90828 

logr/  =  3. 53627  38827  92Sl6 
comp  log  /•'  ='6.46372  61172  07 1 84  =  log  sin  i' 

log;-"  =  5.31442   51331   76459 
comp  log  r"  —  4.68557  48668  23541  =  log  sin  i" 

Let  a  be  the  length  of  an  arc  of  a  circle  whose  radius  is  i , 
and  a"  the  number  of  seconds  in  that  arc,  as 

r1'  —  -J: and  R  :  r"  : :  a  :  a"  or  a"  =  r"  a;  a  =  a"  sin  i" 

sin  i" 

In  an  equation,  therefore,  any  arc  a  of  a  circle  whose  radius  is 
i  is  expressed  in  seconds  by  changing  a  into  a"  sin  \" . 

Signs  of  Trigonometrical  Lines. 

Quadrants.  Sin.      Cos.      Tan.        Cot.     Sec.      Cosec. 

1,  Si  •  9.          C  +      +      +      +      +      +  j 

2,  6,     10,  N    +  -          +    ( 

3»  7»  "»          )  -      -      +      4-  -  C 

4,  8,  12,  &c.  C  -      +      -      -      +      -    > 


10 


WEIGHTS    AND    MEASURES. 


VII. —  Weights  and  Measures  of  the  United  States. 

The  standards  of  length  and  weight  of  this  country  and  Great 
Britain  are  theoretically  identical.  The  United  States  gallon  and 
bushel  represent  old  English  measures. 

The  standard  of  linear  dimensions,  adopted  by  the  Treasury 
Department  in  the  construction  of  standards  for  distribution  to 
the  custom-houses  and  States,  is  a  brass  scale  of  82  inches  in 
length,  made  in  London  by  Troughton,  which  formed  part  of  the 
instruments  collected  in  1815  by  Mr.  Hassler  for  the  Survey  of  the 
Coast,  and  was  supposed  identical  with  the  Schuckburg  scale,  one 
of  the  old  English  standards.  The  standard  temperature  is  62° 
Fahrenheit,  and  the  yard-measure  is  between  the  271)1  and  63d 
inches  of  its  scale.  This  length  has  not  been  legalized  by  act  of 
Congress.  (See  XXXIV.) 


Linear  Measure. 


The  unit  of  linear  measure  is  the  yard.  The  yard  is  divided 
into  3  feet,  and  the  foot  subdivided  into  1 2  inches.  The  multiples  of 
the  yard  are  \hzpole  or  perch,  \hefurlong,  and  the  mile;  but  the  pole 
and  furlong  are  now  scarcely  ever  used,  itinerary  distances  being 
reckoned  in  miles  and  yards. 

The  following  are  the  relations  : 


Inches. 

Feet. 

Yards. 

Poles. 

P\irlongs.              Miles. 

, 

0.083 

0.028 

o.  00505 

o.  00012626 

0.0000157828 

12 

i. 

°-333 

o.  06060 

0.00151515 

o.  00018939 

36 

3- 

i. 

o.  1818 

o.  004545 

0.00056818 

I98 

16.5 

5-5 

I. 

o.  025 

0.003125 

7920 

660. 

220. 

40. 

i. 

o.  125 

63360 

5280. 

1760. 

320. 

8. 

I. 

log  5280  =  3.7226339 
log  1760  =  3.2455127 


WEIGHTS    AND    MEASURES. 


I  I 


VII. —  Weights  and  Measures  of  the  United  States— Continued. 
Square  Measure. 

In  square  measure  the  yard  is  subdivided,  as  in  general  measure, 
into  feet  and  inches ;  144  square  inches  being  equal  to  a  square 
foot.  For  land-measure  the  multiples  of  the  yard  are  the  pole, 
the  rood,  and  the  acre.  Very  large  surfaces,  as  of  whole  countries, 
are  expressed  in  square  miles. 

The  following  are  the  relations  of  square  measure : 


Sq.  feet. 

Sq.  yards. 

Poles. 

Roods. 

i. 

O.  IIII 

0.00367309 

0.000091827 

9- 

'• 

0.0330579 

0.000826448 

272.25 

30.25 

i- 

0.025 

10890. 

I2IO. 

4o. 

*• 

,43500. 

4840. 

TOO. 

4- 

27878400. 

3097600. 

102400. 

2560. 

Measure  of  Capacity. 

The  units  of  capacity  measure  are  the  gallon  for  liquid  and  the 
bushel  for  dry  measure.  '  The  gallon  is  a  vessel  containing  58372.2 
grains  (8.3389  pounds  avoirdupois)  of  the  standard  pound  of  dis 
tilled  water,  at  the  temperature  of  maximum  density  of  water, 
*the  vessel  being  weighed  in  air  in  which  the  barometer  is  30 
inches  at  62°  Fahrenheit.  The  bushel  is  a  measure  containing 
543391.89  standard  grains  (77.6274  pounds  avoirdupois)  of  dis 
tilled  water,  at  the  temperature  of  maximum  density  of  water, 
and  barometer  30  inches  at  62°  Fahrenheit. 

The  gallon  is  thus  the  wine-gallon,  (of  231  cubic  inches,) 
nearly;  and  the  bushel,  the  Winchester  bushel,  nearly. 

The  temperature  of  maximum  density  of  water  was  determined 
by  Mr.  Hassler  to  be  39°.83  Fahrenheit. 


Acres. 

Sq.  miles. 

0.000022957  ! 

O.OOO2O66l2     ; 

0.00625 

0.25 

I. 

640. 

i  . 

DRY   MEASURES. 

LIQUIDS. 

! 

Pint 

=  F4- 

bushel. 

Gill 

=    ^gall. 

Quart 

=  2  pints             =  T,1^ 

bushel. 

Pint 

=    4    gills 

=      i  gall. 

Peck 

=  8  quarts            =  J 

bushel. 

Quart 

=    2    pints 

-      igall. 

Bushel 

=  4  pecks            =   i 

bushel. 

Gallon 

=    4    quarts 

=     i    gall. 

Barrel 

=  3ii  gallons 

=  3ii  galls. 

Hhd. 

=    2    barrels 

=  63    galls. 

The 

only  legalized  unit  of  weight  or 

measure  is  a 

troy-pound, 

, 

I2  WEIGHTS    AND    MEASURES. 


Mil.— Weights  and  Measures  of  the  United  States— Continued. 

(act  of  May  19,  1828,)  copied  by  Captain  Kater,  in  1827,  from 
the  imperial  troy-pound  of  England,  for  the  use  of  the  Mint  of 
the  United  States,  and  there  deposited.  This  pound  is  a  standard 
at  30  inches  of  the  barometer  and  62°  of  the  Fahrenheit  ther 
mometer. 

The  standard  avoirdupois-pound,  as  determined  by  Mr.  Hassler, 
is  the  weight  of  27.7015  cubic  inches  of  distilled  water.  It  is 
greater  than  the  troy-pound  in  the  proportion  of  7000  to  5760; 
that  is,  the  avoirdupois-pound  is  equivalent  in  weight  to  7000 

grains  troy. 

Weights. 

AVOIRDUPOIS.  TROY. 

Dram  =    ^ lb-     i    Grain  =  7Tinr lb- 

Ounce  =i6drs.    =     ^  lb.      |  Pennyweight  =  20  grs.    ,,    ^  lb. 


Pound  =  1  6  ozs.  =  I  lb. 

Quarter  =  25  Ibs.  =      28  Ibs. 

Hundred-wt.  =    4  qrs.  ==    1  12  Ibs. 

Ton  —  20  cwt.  =  2240  Ibs. 

Short  ton  =  2000  Ibs. 


Ounce  —  24  dwt.   =      TV 

Pound  =  12  ozs.    ==        i  Ib. 


VIII  .  —  Miscellaneous. 

/.—  Gunter's  chain  =  66  feet  =  4  poles  =  100  links  of  7.92' 

inches. 

i  fathom  =  6  feet;  i  cable-length  =  120  fathoms. 
i   hand  =  4  inches;    i    palm  =  3  inches;    i  span  =  9 

inches. 
Solid.  —  i  cubic  foot  =  1728  cubic  inches. 

i  cubic  yard  =  27  cubic  feet  =  46656  cubic  inches. 
i  reduced  foot  (board-measure)  =  i   square  foot   X  i 

inch  thick  =  144  cubic  inches. 

i  perch  of  masonry  =  i  perch  (i6J  feet)  long  X  i  foot 
high  x  ij  foot  thick  =  24.75  cubic  feet^  25  cubic 
feet  has  generally  been  adopted  for  convenience. 
i  cord  fire-wood  =  8  feet  long  X  4  feet  high  X  4  feet 

deep  =  128  cubic  feet. 

i  chaldron  coal  =  36  bushels  =  57.25  cubic  feet. 
Paper.  —  24  sheets  =  i  quire. 

20  quires  =  i  ream  =  480  sheets. 


WEIGHTS    AND    VOLUMES. 


IX. —  Weights  and  Volumes  of  various  Substances. 

METALS. 


Substances. 

Cubic  foot. 

Cubic  inch. 

f  Copper..  ..67  i 

Pounds. 

488.  75 

Pounds. 
.2829 

f  Zinc            ."U  \ 

54?.  7? 

.  3147 

547.  25 

.3179 

543.  625 

.3167 

450.437 

.2607 

466.  5 

.  27 

wrought  bars              .*      -   .    ........... 

486.  75 

.2816 

709.  5 

.4106 

rolled               

711.  75 

.4119 

848.  7487 

.  491174 

Steel   plates              -              .................. 

487.  75 

.2823 

soft 

489.  562 

.2833 

Tin                                                      

455.  687 

.  2637 

Zinc   cast                             ..                ........... 

428.812 

.2482 

rolled                .'  

449.437 

.2601 

WOODS. 

Substances". 

Cubic  foot. 

Cubic  feet  in 
a  ton. 

Ash                   .                      

Pounds. 
52.812 

42.  414 

Cedar 

'K.  O62 

63.  886 

Chestnut                                       .      •    

38.  125 

58.  754 

49.  5 

45'  252 

shell-bark 

41.  I2S 

51.  942 

Lignum-  vitcC 

8l.  ^12 

26.  886 

$       35- 

64- 

Oak    Canadian                    .                            ...... 

\       66.437 

C4.  c 

33-  7H 
41.  101 

English           ......  ......  ......  ....  .... 

58.25 

38.455 

live    seasoned 

66.75 

11.  ">S8 

white   dry                      ..............    ... 

ci.  71; 

41.  674 

upland  .......  ......  ......  ...... 

42.937 

52.  169 

WEIGHTS    AND    VOLUMES. 


IX. —  Wrights  and  Volumes  of  "carious  Substances — Continued. 
WOODS — Continued. 


Substances. 


Cubic  foot. 


Cubic  feet  in 
a  ton. 


Pine,  yellow 

Spruce  

Walnut,  black,  dry 
Willow,  dry 


Pounds. 
33-812 
3I-25 
3r-25 
30-  375 


66.  248 
71.68 
71.68 
73-744 


MISCELLANEOUS. 


Substances. 


Cubic  foot. 


Cubic  feet  in 
'a  ton. 


Pounds. 

Air ;  .  07529  r  ! 

Brick,  fire ;     137.562  16.284 

mean i     102.  21.961 

Coal,  anthracite  ..                                     \    ^ "  24'f8 

i  )   102.  5  21.854 

bituminous,  mean 1       80.  28. 

cannel •       94.  875  23.  609 

Cumberland :       84.687  26.451 

Coke |       62.5  35.84 

Cotton,  bale,  mean ^       14.  5  1 54. 48 

!^      20.  114. 

pressed <  _     , 

(    25.  89.6 

Earth,  clay j     120.625  18.569 

common  soil |     137.125  16.  335 

gravel j     109.312  20.49 

dry  sand •. ;      120.  18.667 

lo<>se '      93-75  23.893 

Granite,  Quincy !     165.75      „          ^-SH 

Susquehanna ',     169.  13.  254 

Limestone j     197.25  H'355 

Marble,  mean 167.875  13. 343 

Mortar,  dry,  mean 97. 98  22. 862 

Water,  fresh 62.5  35-84 

salt !       64.125  34. 931 

Steam .036747! 


ARMY-RATION. 


X. — The  Army -Ration. 


TABLE   SHOWING  THE   WEIGHT   AND     BULK   OF     IOOO   RATIONS. 


One  thousand 
rations  of  — 

Net  weight. 

Gross  weight. 

Bulk. 

100  rations  consist 
of— 

Pork  

Pounds, 
750. 
750. 
IMS- 

750- 
IOOO. 

155. 

100. 
100. 

80. 
150. 
92.5 
15. 
40.  " 

33.75 

Pounds. 
1218.75 
903.  19 
1234.06 
921.69 
1228.91 
177.32 
114.50 

122. 

108. 
161. 

107.  50 

17.5° 
46.89 

38.63 

Barrels. 

3-75 
4.90 

5-74 
9.03 
12.05 
0.71 
0.46 
0.65 
0.83 
0.6 

0-33 
0.09 
o.  19 
o.  16 

75  Ibs.  or  i 
75  Ibs.       } 
112.5  Ibs.  or 
75  Ibs.  or 
I  oo  Ibs.  in  the  field 
8  quarts,  or  | 
10  Ibs.           ) 
10  Ibs. 
8  Ibs. 
15  Ibs. 
4  quarts. 
i|  Ibs. 
4  Ibs. 
2  quarts. 

1 

.1 

Bacon  

Flour  

Pilot-bread  
Do 

Beans 

Rice 

Coffee,  green  .. 
roasted. 
Sugar  ..... 

Vinegar  

Candles..     .    . 

Soap  .. 

Salt  

Forage. 

1 4  Ibs.  hay  or  fodder)        ,          (hay,  when  pressed,  1 1  Ibs.  to  cub.  ft. 
/  per  horse  \     *\  c 

12  quarts  oats,  or   >  ^32  Ibs.  to  bushel,  25.71  to  cub.  ft. 

8  quarts  corn         )  pei     ay  (56  Ibs.  to  bushel,  45.02  to  cub.  ft. 
Three  beeves  or  15  sheep  consume  the  forage  of  2  horses. 


Wheat , 60  Ibs.  | 

Corn  and  rye 56  Ibs.  J 


Weights  of  Grain  per  Bushel. 

Oats  ....... 

Barley 


32  Ibs. 
48  Ibs. 


A  box  1 6  x  1 6. 8  x  8.  inches  contains  i  bushel  \ 

12  x  1 1. 2  X  8.  "  £  bushel  >  dry  measure. 

8  x     8.4  x  8.  "  i  peck     ) 

6  X        6  x  6.4  "  i  gallon 

4  X        4  X  3.6  "  i  quart 


i6 


METRIC    SYSTEM. 


XI. — Metric  System. 

By  an  act  of  Congress,  approved  July  28,  1866,  the  metric 
system  of  weights  and  measures  is  made  optional  in  the  United 
States;  and  the  act  provides  that  the  tables  in  a  schedule  an 
nexed  shall  be  recognized  "as  establishing,  in  terms  of  the  weights 
and  measures  now  in  use  in-  the  United  States,  the  equivalents  of 
the  weights  and  measures  expressed  therein  in  terms  of  the  metric 
system;  and  said  tables  may  be  lawfully  used  for  computing,  de 
termining,  and  expressing,  in  customary  weights  and  measures, 
the  weights  and  measures  of  the  metric  system." 

Schedule  annexed  to  act  of  July  28,  1866. 
MEASURES  OF  LENGTH. 


Metric  denominations. 

Values  in  metres. 

Equivalents  in  denominations  in  use. 

Myriametre               .   ... 

6.2137  niiles*. 

Kilometre  

IOOO. 

0.62137  mile,  or  3280  feet  and  10  inches. 

Hectometre 

,00. 

328  feet  and  i  inch 

Decametre  
Metre 

':: 

393.7  inches. 
39.37  inches. 

Decimetre  

O.I 

3.937  inches. 

Centimetre 

Millimetre  

MEASURES  OF  SURFACE. 


Metric  denominations. 

Values  in 
square  metres. 

Equivalents  in  denominations  in  use. 

Hectare  

IOOOO 

2.471  acres. 

Are         

119  6  square  yards. 

Centare 

MEASURES  OF  CAPACITY. 


Metric  denominations  and  values. 

Equivalents  in  denominations  in  use. 

Names. 

No.  of 

litres. 

Cubic  measure. 

Dry  measure. 

Liquid  or  wine 
•      measure. 

Kilolitre  or  stere. 
Hectolitre  
Decalitre  

IOOO. 
IOO. 
IO. 

O.  I 
0.01 
O.OOI 

i  cubic  metre  
o.i  cubic  metre 
10  cubic  decimetres 
i  cubic  decimetre  .  . 
o.i  cubic  decimetre 
10  cubic  centimetres 
i  cubic  centimetre  . 

1.308  cubic  yards  . 
2  bus.  and  3.3spks. 
9  08  quarts  

264.17  gallons. 
26.417  gallons. 
2.6417  gallons. 
1.0567  quarts. 
0.845  gill. 
0.338  fluid-ounce. 
0.27  fluid-drachm. 

Litre 

0.908  quart  
6.1022  cubic  inches 
0.6102  cubic  inch.  . 
0.061  cubic  inch.  .. 

Decilitre  

Centilitre 

Millilitre.    . 

METRIC  SYSTEM. 


XI . — Metric  System — Continued. 

WEIGHTS. 


Metric  denominations  and  values. 

Equivalents  in  denom 
inations  in  use. 

Names. 

Number  of 
grammes^ 

Weight  of  what  quantity  of 
water  at  maximum  density. 

Avoirdupois  weight. 

Millieror  tonneau.  . 
Quintal  
Myriagramme  
Kilogramme,  or  kilo 
Hectogramme  ...... 

1000000. 
IOOOOO. 
IOOOO. 
IOOO. 

i  cubic  metre  

2204.6  pounds. 
220  46  pounds. 
22.046  pounds. 
2.2046  pounds. 
3.5274  ounces. 
0.3527  ounce. 
15.432  grains. 
1.5432  grains. 
0.1543  grain. 
0.0154  grain. 

i  hectolitre  
10  litres  
i  litre  

i  decilitre 

Decagramme  
Gramme  
Decigramme  
Centigramme  
Milligramme  

10. 

I  . 

O.  I 
0.01 
O^OOI 

10  cubic  centimetres  

i  cubic  centimetre  . 

o.  i  cubic  centimetre  
10  cubic  millimetres 

i  cubic  millimetre.  .  .  . 

ADDITIONAL    METRICAL   KQUIVALEXTS. 

I  surveyor's  chain  in  metres..   =:     20.11662     log  =  1.3035550 

i  metre  in  surveyor's  chain  ..   =       0.04971      log  — •  8.6964450 


I  square  foot  in  square  metres  =       0.09290     log  • —  8.9680221 

I  acre  in  hectares =         0.40467     log  =  9.  6071100 

I  square  mile  in  hectares =  258.994         log  =  2.4132900 


I  square  metre  in  square  feet. 

I  hectare  in  acres 

I  hectare  in  square  miles 


I0-  764io     log  =  1.0319779 

2.  47i09     log  —  o.  3928900 

0.00386     log  =  7.5867100 


I  cubic  foot  in  sfceres =       o.  0283 1     log  =  8.  4520332 

I  cord  in  steres =       3.62445     log  =  o.  5592432 


i  stere  in  cubic  feet =    35.31561     log  =  i.  5479668 

I  stere  in  cords =       0.27590.    log  =  9.  4407568 


I  gram  in  grammes 


=      0.064798  log  =  8.  8115680 


18 


FOREIGN    MEASURES. 


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co  'o     co   t^-      ro  ir»      "^r  t>^     *O   vN       O>  "^                   o   M 

o 
0 

C    II 

r0    "•> 

. 

§8 

F 

cJJ  O 

3  "M 

cj    O*         O    O        MO'         M"    O*         CN|    o'                           M"    o'         MO 

•i 

•ii 

t^»co         MosOn'VOvo                            C^^t-rJ-o         ON 
CN.  o'         MO'         t^.  o'         !>•  o'                            O    ON      O    o'         l-O  o' 

rt 

1  1 

CO  o)        ^"  ro       O^  -o                      co  ^-       cOco        ^^  ^-       O  vo 

< 

MO       Oooo                     Oooooooo 

y.  s 

?  ^ 
PI  i~"t 

4 

1  1!  -  1  !!  I!  1  1 

To 

it 

II  -  II  I!  It  1!  II  IS 

H 

"rt    || 

C/5 

VO    O                          r^~  0         T}~  O         O    O       O    0         Tj"  O         CO  O 

8 

rt 

Myriametre 
—  i  oooo  M. 

to            i-O             CO             M              M             CO             CO 

MO       COro^-Coo       CO«         cOo        tj-i  M         M    ^j. 
CO  ^_       ^co^      O^       t^toCOoo         Mro       ^4: 

HH     04           1^.00           i>.CO           M     O           ^CJ"O           t^.CO           \O   ^ 

do     do     do     do     do     do     do 

Eng.  stat.  miles,  jj  Eng.  stat.  miles. 

Modern  Roman  mile =    0.925    j  Portugal  league =    3.841 

Tuscan  mile =     1.027  <    Flanders  league =     3.  900 


Old  Scottish  mile 
Irish  mile . ., 


=     1.127 
=     1-273 


Spanish  common  league  .  =    4.  214 


Hungarian  mile. 


French  posting  league  ..   --=     2.422  |    S\yedish  mile 


-    5.178 
=    6.648 


20 


COMPARATIVE    MEASURES. 


Table  for  converting  Metres  into  Toises  and  French  and  English 
Feet  and  Inches. 


Metres. 

Toises. 

French. 

English. 

Feet. 

Inches. 

Lines. 

Feet. 

inches. 

I 

0.51307 

3 

o 

11.296 

3 

3.3708 

2 

1.02615 

6 

i 

10.592 

6 

6.  7416 

3 

1.53922 

9 

2 

9.888 

9 

10.  1124 

4 

2.  05230 

12 

3 

9.  184 

13 

1.4832 

2.  56537 

15 

4 

8.480 

10 

4-  8539 

6 

3.  07844 

18 

5 

7.776 

19 

8.  2247 

7 

3-59I52 

21 

6 

7.072 

22 

11-5955 

8 

4-  10459 

24 

7 

6.368 

.  26 

2.  9653 

9 

4.61767 

27 

8 

5.664 

29 

6-337* 

10 

5-  T3°74 

30 

9 

4.960 

32  1    9-  7079 

20 

10,  26148 

61 

6 

9.920 

65  !   7-4158 

30 

15.39222 

92 

4 

2.880 

98    5-  I237 

40 

20.  52296 

123 

i 

7.840 

131     2.8316 

25.65370 

153 

ii 

o.  800 

164 

o.  5395 

60 

30.  78444 

184 

8 

5-  /^o 

196 

10.2474 

70 
80 

35-9I5J9 
4i'°4593 

2I5 
246 

5 
3 

10.  720 
3.680 

229 
262 

7-9553 
5.6632 

90 

46.  17667 

277 

0 

8.  640 

295    3-3/11 

100 

-0741 

3°7 

10 

i  600 

328     1.0790 

200 

102.61481 

615 

8 

3.  200 

656       2.  1580 

300 

153.92222 

923 

6 

4.  800 

984 

3-2370 

400 

205.  22963 

1231 

4 

6.  400 

I3I2 

4.3160 

600 

256.  53704 
307.  84444 

1539 

1847 

o 

8.000 
9.  600 

1640       5.3950 
1968       6.4740 

700 
800 

359.15^85 
410.45926 

2154 
•   2462 

10 

9 

!  1  1.  200 
o.  800 

2296       7.  5530 

2624     8.  6320 

900 

46  1.  .76667 

2770 

7 

2.  400 

2952     9.7110 

IOCO 

513.07407 

3078 

5 

4.  oco 

3280    10.  7900 

20CO 

1026.  14815 

6156 

10 

,  8.  O;)0 

6561 

9.  5800 

3000 

1539.22222 

9235 

4 

o.  ooo 

9842 

8.  3700 

4000 

2052.  29630 

12313 

9 

4.  ooo 

i3I23 

7.  1600 

5000 

2565.37037 

15392 

2 

8.000 

16404 

5-  95°o 

6000 

3078.  44444 

18470 

8 

o.  ooo 

19685 

4.  7400 

7000 

3591  51852 

21549 

i 

4.  ooo 

22966 

3-  53°° 

8f»oo 

4104.  5925.9 

24627 

6 

8.000 

26247 

2.  32OO 

9000 

4617.  66667 

27706 

0 

0.  000 

29528     i.  i  100 

IOOOO 

5130.  74074 

30784 

5 

4.000 

-  | 
32808  i   11.9000 

COMPARATIVE    MEASURES.                                               21 

Table  for  converting  English  Feet  into   French    Toises,   Metres, 
and  Feet. 

English 
feet. 

Toises. 

Metres. 

Ffench  . 

Feet. 

. 

Inches.           Lines. 

2 

O 

o.  15638 

0.31276 
0.46915 

o.  30479 
o.  60959 

0.91438 

o 

2 

II              3.114 

10  ;          6.  228 
9            9-  343 

4 
6 

0.62553 

o.  78191 
o.  93829 

I.  21918 

1.52397 
1.  82877 

3 
4 
5 

9             0.457 
8             3-571 
7            6.  685 

8 
9 

I.  09468 
I.  25106 

1.40744 

2.  13356 
2.43836 

2.  743  i  5 

6                  69.  799 
7  !               6            0.913 
8                 5  -          4.028 

10 

20 
30 

1.56382 

3-  12764 
4.  69146 

3-  °4794 
6.  09589 

9-  H383 

9  :            4  j        7.  142 
18  :           9  ;       2.284 
28  ;              i           9.425 

40 

£ 

6.  25529 
7.  81911 
9-  38293 

12.  19178 

15-23972 
18.28767 

37                  6  !          4.567 
46                10          ii.  709 
5^J  '               3  I          6.851 

£ 

90 

10.  94675 

12.  5105? 
14-  °7439 

2i.3356i 
24-  38536 
27-4315° 

65  i               8             L993 
75                  o  i          9.  134 
84  ;               5            4.276 

IOO 
200 
3OO 

15.63822 

31-  27643 

46.91465 

3°.  47945 
60.  95850 

9L43835 

93                  9  !        11.418 
187  ;              7          10.  836 
281                 5  ;        10.254 

40O 

c;oo 
600 

62.  55286 
78.  I9I08 
93.  82929 

121.  91780 

152.39725 
182.87670 

375                 3            9-672 
469                 i  ,          9.090 
562                ii            8.508 

700 
800 
900 

109.46751 
125.10572 

140.  74394 

2I3.356I5 
243-  83559 
274.31504 

656 
750 

844 

9             7.  926 
7             7-344 
5            6.  762 

IOOO 
20OO 

3000 

156.38215 
312.76431 
469.  14646 

304.  79449 
609.  58899 
914.38348 

938 
1876 
2814 

3            6.  i  So 
7            o.  360 
10            6.539 

4000 
5000 
6000 

625.52861 
781.  91076 
938.  29292 

1219.17797 
1523.  97246 
1828.  76696 

3753 
4691 
5629 

2            o.  719 
5            6-  899 
9             I«°79 

7000 

8000 

9000 

1094.  67507 
1251.  05722 
1407.43937 

2i33.56i45 
2438.  35594 
2743-  !5044 

6568 

75°6 
8444 

o             7.  259 
4             1.438 
7             7.618 

IOOOO 

1563-82153 

3047.  94493 

9382 

ii             1.798 

0  2 

MENSURATION. 

j  , 

XIV.  —  Analytical  Expressions  for  different 

Lines,  Surfaces,  and 

Solids. 

i.  —  LINES. 

Ratio 

of  diagonal  to  side  of  square  =  \/  2  - 

=  I-4H  = 

Y-,  nearly. 

Log  i 

/2    =  0. 

15°5I49978 

Side  of  inscribed  square  :  R  :  :  \?  2  :  i 

Side  of  inscribed  equilateral  triangle  :  R  :  : 

\/3  :  i 

Side  of  inscribed  regular  hexagon  =  R 

Side  of  inscribed  regular  decagon  =  J  R  (- 

-  1  +  Vs) 

=  0.618  R 

Circle. 

Ratio 

of  circumference  to  diameter  =  3.1415926  =  f^ 

g,  nearly. 

Length  of  an 

arc  : 

=  'llo:'    r   be!ng   the 

radius   of 

the 

circle 

and  a  the  number  of  degrees 

in  the  arc  ; 

or  nearly  = 

_  8  c1  —  c 

c  being  the 

chord  of  the  arc, 

and  c1  (the 

chord  of  half  the  arc) 

=  V  i  <?  + 

ver  sin2 

Ellipse. 

ly;  a  and  b  being  the 

Circumference  =  ~ 

OS-"  V4  («a 

+  I}-}  near 

axes. 

Lengths  of  Circular  Arcs,  taking  the  Base  of  Segments  as  Unity. 

"x 

"33 

ti            'p5 

t^                     c 

^° 

"~Ti 

i 

I* 

t° 

o 

-1 

1 

oj          '     ^5 

0                     J 

I-H                       > 

3            3  • 

1 

J 

.  OI 

I.  OOO 

.  II 

.032        .21 

1.114    .; 

51        1.239 

.41 

1.401 

.  O2 

I.  000 

.  12 

.  038    i     .  22 

1.124     : 

52        1.254 

.42 

1.418 

.03 

I.  OOO 

.13 

.044   i     .23 

1-135    •: 

53        1-269 

•  43 

1.437 

.04 

I.  OOO 

.  14 

.051        .24 

1.147    •; 

54       1-284 

•  44 

1.455 

.05 

I.  000 

.15 

•°59      -25 

1-159    •: 

55       i-3°° 

.45 

1.474 

.06 

1.  006 

.16 

.  067  i   .  26 

1.171     .; 

}6       1.316 

.46 

1-493 

.07 

1.014 

•17 

.075  !   .27 

1.184     .; 

57       L332 

•  47 

1.512 

.08 

1.018 

.18 

.  084  ;!  .  28 

1-197    •; 

58       1.349 

.48 

I-53I 

.09 

1.020 

.19 

.093     .29 

i.  212      .; 

59       i-366 

•49 

J.55i 

.  10 

I.  026 

.  2O 

.103     .  30 

1.225      .< 

10       1.383 

.50 

I-57I 

MENSURATION.  23 


XIV. — Analytical  Expressions,  &c. — Continued. 

2. — SURFACES. 

• 

i.  Triangle  in  terms  of — 

b  A 
its  base  and  its  altitude =  - 

2 

a  b  sin  C 
two  sides,  and  the  included  angle = 

its  three  sides =  [s  (s  —  a)  (s  —  b}  (s  —  c} }  * 

where 

A  =  the  altitude; 
(7,  b,  c  =  the  three  sides; 

C  =  the  angle  included  between  a  and  b ;  and 
a  +  b  +  c 


2.  Parallelogram  in  terms  of  — 
its  base  and  its  altitude     ..............    =  b  A 

two  sides  and  the  included  angle    ........  =  a  b  sin  C 

two  sides  and  their  corresponding  diagonal 

=  2  [s(s-a)(s-t)(s-c)]* 
where 

C  =  the  angle  included  between  two  adjacent  sides  a,  b  ; 
<r=the  diagonal  opposite;  and 


2 

3.  Trapezium  in  terms  of  — 

its  two  parallel  bases  and  its  altitude  .......    =—      -A 

its  two  parallel  bases,  one  of  its  oblique  sides,  and  the  angle 
between  one  of  these  bases  and  this  side    .    .    .    =  —     —  /  sin  C 

2 

where 

A  —  the  distance  between  the  two  parallel  bases  B,  b; 
7=  the  length  of  one  of  the  oblique  sides;  and 
C  =  the  angle  between  one  of  these  bases  and  this  side. 

4.  Any  quadrilateral  =  half  the  product  of  its  two  diagonals 

multiplied  by  the  sine  of  the  included  angle. 


MENSURATION. 


XIV. — Analytical  Expressions,  &c. — Continued. 


5.  Regular  polygon 
where 


tan 


i8oc 


n  =  the  number  of  sides;   and 
a  =  the  length  of  one  of  them. 

6.  Circle =  -  R2 

7.  Ellipse =  -  a  I) 

a  and  b  being  the  semi-axes. 

8.  Right  cylinder,  exclusive  of  its  bases    .    .    .    .    =  2  -  R  A 

9.  Sphere =  4  -  R2 

10.  Zone =  4  -  R2  sin  -J  (L'  —  L)  cos  A  (L7  +  I,) 

1 1 .  Right  cone =  -  R  L 

12.  Frustum  of  cone  with  parallel  bases    .    .    .    =  -  /  (R  -f-  '") 
where 

R  and  r  —  the  radii  of  the  bases  of  these  solids;  and 
L  and  /  =  the  lengths  of  their  generating  elements. 

13.  Spherical  quadrilateral,  formed  by  two  parallels  of  latitude 
and  two  meridians 


=  -Qo  (M7  -  M)  R2  sin  A  (L7  -  L)  cos  -J  (L7  +  L) 


where 


R  =  the  radius  of  the  sphere; 
L,  L7  =  the  latitudes  of  the  bases  of  the  zone,  +  when 

north,  —  when  south;  and 

INI7,  M  =  the  longitudes  of  the  extreme  meridians  of  the 
quadrilateral,   (M7  —  M)   being    expressed  in 
degrees  and  decimals. 
In    the   place    of  R,   the    normal    N,   of  the    mean    latitude 

,  can  be  used. 


MENSURATION. 


25 


XIV.  —  Analytical  Expressions,  &c.  —  Continued. 
3.  —  SOLIDS. 

14.  Prism  ...........    ........    =  B  A 

where 

B  =  the  area  of  the  base;  and 
A  =  the  altitude. 

15.  Rectangular  parallelopepidon    ......    =  p  x  q  X  1 

Cube    ................    =/3 

where/,  </,  r,  =  the  lengths  of  the  three  contiguous  edges. 

"R     A 

1  6.  Pyramid  ................    =  ------ 

The  area,  B,  being  found  from  Xo.  5. 

17.  Right  cylinder    .............    =  -  R2  A 

18.  Right  cone     ..............    =  £  -  R2  A 

19.  Sphere     ................    =  A  -  R3 

20.  Prismoid,  or  solid  figure,  similar  to  that  which  is  formed 
in  excavations  or  embankments  of  roads,  terminated  by  parallel 
cross  -sections. 

Solid  content  =  area  of  each  end,  added  to  four  times  the 
middle  area,  and  the  sum  multiplied  by  the  length  divided 
by  6,  or 


where 

b  =  the  breadth  at  the  bottom  of  the  cutting  ; 
h  =  the  perpendicular  depth  of  cutting  at  higher  end; 
h1  •=.  the  perpendicular  depth  of  cutting  at  lower  end; 
/=  the  length  of  the  solid;  and 

r  =  the  ratio  of  the  perpendicular  height  ot   the  slope  to 
its  horizontal  base. 


26  MISCELLANEOUS. 


X  V .  — Progression . 

1.  Arithmetical: 

a  =  z  —  ( 11  —  i  )  d        z  =  a  +  (n  —  i)d 

z  —  a  z  —  (?    ,  a  -4-  z 

d  =  -  n  =  —  ,  —  +  i          s  = 

n  —  i  d  2 

2,  Geometrical : 


z  =  at"1 


n  =  log.  .££.  H.  1OI!. 


a  =  —s          a  =  z  r  —  (r  —  i )  s 

r'1  —  i  .$•  —  z 

where 

#  =  the  least  term;  ;-  =  the  common  ratio; 

z  =  the  greatest  term;  n  =  the  number  of  terms; 

d  =  the  common  cliff. ;  s  =  the  sum  of  the  terms. 

XVI. — Force  of  Gravity. 

The  velocity  acquired  at  the  end  of  one  second  by  a  body 
falling  in  vacuo,  at  the  level  of  the  sea,  in  the  latitude  of  Lon 
don  =  32.1915  feet. 

The  force  of  gravity  at  the  latitude  of  45°  =  32.17  feet  per 
second  being  represented  by  g,  for  any  other  latitude,  /, 
g'  =  g  (j  _  0.002588  cos  2  /) 

If  ^represents  the  force  of  gravity  at  the  height  //,  and  r  the 
radius  of  the  earth,  the  force  of  gravity  at  the  level  of  the  sea 


Length,  in  inches,  of  a   pendulum  vibrating   seconds   at  the 
level  of  the  sea  : 

Equator      ....  =39.0152  I1  London,  lat.  51°  31'   -      -  —  39.1393 

New  York,  lat.  40°  43'  =39.1017  [|  Spitzbergen,  lat.  75°  50'     =39.2147 


LAND-SURVEYING. 


XVII. — Land- Survey  ing  with  Compass  and  Chain. 
2o  calculate  the  Area  or  Content  of  Land. 

If  the  sum  of  each  adjacent  pair  of  distances  perpendicular  to 
a  meridian  (departures]  assumed  without  the  survey  be  multi 
plied  by  the  northing  or  southing  between  them  in  succession 
round  the  figure  in  the  same  order,  the  difference  between  the 
sum  of  the  north  products  and  the  sum  of  the  south  products 
will  be  double  the  area  of  the  tract. 

The  meridian  distance  of  a  course  is  the  distance  of  the  middle 
point  of  that  course  from  an  assumed  meridian. 

Hence,  the  double  meridian  distance  of  the  first  course  is 
equal  to  its  departure. 

And  the  double  meridian  distance  of  any  course  is  equal  to 
the  double  meridian  distance  of  the  preceding  course,  plus  its 
departure,  plus  the  departure  of  the  course  itself,  having  regard 
to  the  algebraic  sign  of  each. 

Then,  to  find  the  area — 

1.  Multiply  the  double  meridian  distance  of  each  course  by  its 

northing  or  southing. 

2.  Place  all  the  //^products  in  one  column,  and  all  the  minus 

product's  in  another. 

3.  Add  up  each    column   separately,  and  take  their  difference. 

This  difference  will  be  double  the  area  of  the  land. 
In  balancing  the  work,  the  error  for  each  particular  course  is 
found  by  the  proportion — 

As  the  sum  of  the  courses  is  to  the  error  of  latitude,  (or  de 
parture,  )  so  is  each  particular  course  to  its  correction. 
When  a  bearing  is  due  east  or  west,  the  error  of  latitude  is 
nothing,  and  the  course  must  be  subtracted  from   the  sum  of  the 
courses  before  balancing  the  columns  of  latitude.      And  so  with 
the  departures. 

EXAMPLE. — It  is  required  to  find  the  content  of  a  piece  of  land,  of  which 
the  following  are  the  field-notes  : 


Sta. 

Course. 

Dist. 

Sta. 

Course. 

Dist. 

i 

North  46^-°  west. 

20.       chains. 

4 

South  56°  east  .  . 

27.60  chains. 

2 

North  5if°  east. 

13.80  chains. 

!     5 

South  33^°  west.    18.80  chains. 

3 

East 

21.25  chains. 

I6 

North  74!°  west. 

30.95  chains. 

' 

28 


LAND-SURVEYING. 


'"5 

•^ 
"^* 

XVII.—  Land- 

1    1 

Surveying,  &c\  —  ( 

:    :      ?o£ 

•           •                  OO        LT) 

.     .         t^  o 

Continued. 

o  oo      oo       Tf 

N      ON         N         vO 
T}-   CO           >J->         N 
OO      !>.          O            LO 

C\    N          rA         ro 

ii                t~>.     M 

:    :      £?  ^? 

00      ^        °0          Q" 

i  + 

-S 

00     i-                             ON 
NT)-                                 N 
ON    t^                           <N 

O      O                             O 

N      ro                              1^- 

N                                         ^f 

Q 

<•;    T 
X 

10  OO    CO     OO      CO    C 

ti 

Tr 
0 

?* 

rj-     O      N     v£5      O     CN 

o 

d          s 

»-O  00      N     OO      fO    O 

?  2  N  8  2  £ 
1   +  +  +  1    1 

ri 
'rt                   ^J 

3 

§8  ^    :  ^  ^  3. 

ro  OO        i     vo    uo  00 

+  +  i  1   1  + 

^  i 

^       :      Toe? 

O                 '^ 

^. 

S      o 

t~>.    u^         N 

rf      rf           O*           " 

Tf                •               0      CN 

f  .+ 

Tf    u-i  00        • 
CO      N     00 

0      i-''    N         • 

*     *  ' 

•      T)-     N 
.      TT    t^ 

>O     00         CO         •< 

W        LO            LO 

I     ^    ^ 

r;  d    o    o 

0 
00 

LO                                              i 

6                  < 

O                 t/3 
N                   )H 

1-1             o 

id 

5         .   . 

t~>.     i-O       i         t         i      N 

•ro  00*       •        •        -co' 

c  Jj 

8O     to    O     O     ^O 
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N      1-1      N      N      >-i      fO 

c 
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w 

X 

o"1     !  W  *"* 

5  S  |:  %,  ^  3 

•suoi^S        ~    "-,  ^  o 

IOODOO  square  links  of  Gunter's  chain  —  I  acre. 
I  square  chain  =  65  feet  square  =  iLy-  acre. 


TRAVERSE    TABLE, 


XVIII. —  Table  showing  Differences  of  Latitude  and  Departures. 


45 


0° 

1° 

2° 

QJ 
0 

C 

w 

<D 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  j.2 

M 

C 

I  .  OOOOO 

0.00000 

0.90984 

0.01745 

0.99939 

o.  03490  h 

I 

2  .  OOOOO 

o  .  ooooo 

i  .  99969 

0.03490 

1.99878 

0.06980 

2 

3  .  ooooo 

o  .  ooooo 

2-99954 

0.05235 

2.99817 

O.I04701 

3 

4  .  ooooo 

0.00000 

3-99939 

0.06980 

3.99756 

o  .  i  3960 

4 

5.00000 

o.  ooooo 

4.99923 

0.08726  ' 

4.99695 

0.17450 

5 

60 

6  .  ooooo 

o  .  ooooo 

5.99908 

o.  10471  i 

5-99634 

0.20940 

6 

7  .  ooooo 

o  .  ooooo 

6.99893 

0.12216 

6-99573 

0.24430 

7 

8  .  ooooo 

o  .  ooooo 

7.99878 

0.13961  ' 

7.99512 

0.27920 

8 

g  .  ooooo 

o  .  ooooo 

8.99862 

0.15707 

8.99451  0.31410 

9 

0-99999 

0.00436 

0.99976 

0.02181 

0.99922 

0.03925 

i 

i  .  99998 

0.00872 

1.99952 

0.04363 

1.99845 

0.07851  jl  2 

2.99997 

0.01308 

2.99928 

0.06544 

2.99768 

0.11777 

3 

;  3.99996  0.01745 

3.99904 

0.08725 

3.99691  0.15703!  4 

4.99995  i  o.  0218  1 

4.99881 

o.  10907 

4.99614 

0.19629  5 

45 

5.99994  0.02617 

5.99857 

0.13089 

5-99537 

0.23555  6 

6.99993  10.03054 

6.99833 

0.15270 

6.99460 

0.27481  |  7 

7.99992  0.03490 

7.99809 

o.  17452 

7.99383 

0.31407  8 

8.99991 

0.03926 

8.99785 

0.19633 

8.99306 

0.35333!  9 

0.99996  !  0.00872 

0.99965 

0.02617 

0.99904 

0.04361 

i 

1.99992  0.01745 

1.99931 

0.05235 

1.99809 

0.08723 

2 

2.99988 

0.02617 

2.99897 

0.07853 

2.99714 

0.13085 

3 

3-99984 

0.03490 

3.99862 

o.  10470 

3.99619 

0.17447;  4 

4.99981  j  0.04363 

4.99828 

0.13088 

4.99524 

0.21809  5* 

30 

5.99977  0.05235 

5-99794 

0.15706 

5.99428 

0.26171 

6 

6.99973  0.06108 

6.99760 

0.18323 

6-99333 

0-30533; 

7 

7.99969  0.06981 

7.99725 

0.20941 

7.99238 

0.34895 

8 

8.99965  10.07853 

8.99691 

0.23559 

8.99143 

0.39257 

9 

0.99991 

0.01308 

0-99953 

0.03053 

0.99884 

0.04797   i 

1.99982  0.02617 

i  .  99906 

0.06107 

1.99769 

0.09595 

2 

2.99974 

0.03926 

2.99860 

0.09161 

2.99654 

0.14393  3 

i  3.  9996^ 

0.0^235 

3.99813 

o.  12215 

3-99539 

0.19191  4 

4.99957 

0.06544 

4.99766 

o.  15269 

4.99424 

0.239891!  5 

15 

i  5.99948  0.07853 

5.99720 

0.18323 

5.99309 

0.28786  |  6 

16.99940  0.09162 

6.99673 

0.21376 

6.99193 

0.33584  7 

7-99931 

o.  10471 

7  .  99626 

0.24430 

7.99078 

0.38382 

8 

8.99922 

0.11780 

8.99580 

0.27484 

8.98963 

0.43180 

9 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

O 

g 

P 

c 

89°            88° 

87° 

D 
n 
a 

o 
en 

30                                                     TRAVERSE    TABLE. 

Differences  of  Latitude  and  Departures  —  Continued. 

. 

o 

3° 

4° 

5° 

0 
CJ 

if. 

3 

G 

C 

— 

c 

to 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

uo 

p 

g 

I 

0.99863 

0.05233    0.99756 

0.06975 

0.99619 

0.08715 

I 

2    | 

1.99726 

0.10467 

1.99512 

O.I395I 

1.99238 

0.17431 

2 

3  . 

2.99589 

0.15700 

2.99269 

0.20926' 

2.98858 

0.26146 

3 

4  | 

3.99452 

0.20934 

3.99025 

0.27902 

3.98477 

0.34862 

4 

o 

4.99315 

0.26168 

4.98782 

0.34878 

4.98097 

0-43577 

5 

60 

6 

5.99178 

0.31401 

5.98538 

0.41853 

5.97716 

0.52293 

6 

7 

6.99041 

0.36635 

6.98294 

0.48829 

6.97336 

0.61008 

7 

8  i 

7.98904 

0.41868 

7-98051 

0.55805 

7.96955 

0.69724 

8 

9 

8.98767 

0.47102 

8.97807 

0.62780 

8.96575 

0.78440 

9 

i 

0.99839 

0.05669 

0.99725 

0.07410 

0.09580 

0.09150 

T 

2 

1.99678 

0.11338 

1.99450 

o.  14821 

1.99160 

0.18300 

2 

3 

2.99517 

o  .  1  7.007 

2.99175 

0.22232 

2.98741 

0.27450 

3 

4 

0.22677 

3-98900 

0.29643 

3-98321 

0.36600 

4 

15 

4.99J95 

0.28346 

4.98625 

0.37054 

4.97902 

0.45750 

5* 

45 

6 

5-99035 

0.34015 

5.98350 

0.44465 

5.97482 

0.54900 

b 

7 

6.98874 

0.39684 

6.98075 

0.51875 

j  6.97063 

0.64051 

7 

8 

7-98713 

0.45354 

7.97800 

0.59286 

7-96643 

0.73201 

8 

9 

8.98552 

0.51023  ; 

8.97525 

0.66697 

8  .  96224 

0.82351 

9 

i 

0.99813 

0.06104 

0.99691 

0.07845 

:  0.99539 

0.09584 

i 

2 

1.99626 

0.12209 

1.99383 

0.15691 

}  1.99079 

o.  19169 

2 

3 

2  •  99440 

0.18314 

2.99075 

0.23537 

2.98618 

0.28753 

3 

4 

3-99253 

0.24419 

3-98766 

0.31383    3-98158 

0.38338 

4 

30 

4.99067 

0.30524 

4.98458 

0.39229  14.97698 

0.47922 

5 

30 

6  i 

5.98880 

0.36629 

5.98150 

0.470751!  5.97237 

0.57507 

b 

8 

6.98694 
7.98507 

0.42733 

0.48838 

6.97842 
7-97533 

0.54921    ;  6.96777 
0.62/67    i  7.96316 

0.67092 
0.76676 

8 

9 

8.98321 

0.54943 

8.97225 

0.70613 

8.95856 

K 

0.86261 

9 

i 

9-99785 

0.06540 

0.99656 

0.08280 

0.99496 

o.  10018 

i 

2 

I-9957I 

o.  13080 

L993I3 

0.16561  |    1.98993 

0.20037 

2 

3 

2-99357 

o.  19620 

2.98969 

0.24842  ;'  2.98490 

0.30056 

3 

4 

3.99I43 

0.26161 

5.98626 

0.33123'  3-97987 

0.40075 

4 

45 

4.98929 

0.32701 

4.98282 

0.41404    4.97484 

0.50094 

5 

15 

6 

0.39241 

5-97939 

0.49684    5.96981 

0.60112 

b 

7 

6.98501 

0.45782 

6-97595 

0.57965    6.96477 

0.70131 

7 

8 

7.98287 

0.52322 

7.97252 

0.66246    7-95974 

0.80150 

8 

9 

8.98073 

0.58862 

8  .  96908 

0.74527    8.95471 

0.90169 

9 

g 

3 

O 

on" 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

a 

En 

g 

C 

&3 

P 

c 

.    0 

o 

O 

86°                         .   S5'J                             84°                 P 

c 

7) 

• 

TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures — Continued. 


0> 

6 

7 

8 

1 

6 

0 

o 

e 

c 

o 

— 

rt 

rt 

£3 

G 

i 

en 

s 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

en 
Q 

i 

I 

o  99452 

0.10452 

0.99254 

o.  12186 

0.99026 

0.13917 

2 

1.98904 

0.20905 

1.98509 

0.24373 

1.98053 

0.27834 

2 

•3 

2.98356 

0.31358 

2.97763 

0.36560 

2.97080 

0.41751 

3 

4 

3.97808 

0.41811 

3.97018 

0.48747 

3-96107 

o.  55669 

4 

o 

5 

4.97261 

0.52264 

4.96273 

0.60934 

1  4-  95134 

0.69586 

5 

60 

6 

0.62717 

5.95519 

0.73121 

i  5-  94160 

0.83503' 

6 

7 

6.96165 

0.73169 

6.94782 

0.85308 

,6.93187 

0.97421 

7 

8 

7.95617 

0.83622 

7.94038 

0-97495 

7.92214 

0.11338 

8 

9 

8.95069 

0.94075 

18.93291 

0.09682 

8.91241 

0.25255 

9 

j 

0.99405 

0.10886 

0.99200 

0.12619 

0.98965 

0.14349 

i 

2 

1.98811 

0.21773 

i  .  98400 

0.25239 

1.97930 

0.28698 

2 

^ 

2.98216 

0.32660 

2.97601 

0.37859 

2.96895 

0.43047 

3 

4 

3.97622 

0.43546 

3.96801 

0.50479 

3.95860 

0.57397 

4 

15 

4.97028 

0-54433 

4  .  96002 

0.63099 

4.94825 

0.71746 

5 

45 

6 

5.96433 

0.65320 

5.95202 

0.75719 

5.93790 

0.86095 

6 

7 

6.95839 

0.76206 

6.94403 

0.88339 

6.92755 

1.00444 

7 

8 

7.95245 

0.87093 

7.93603 

1.00959 

7.91721 

I.I4794 

8 

9 

8.94650 

0.97980 

8.92804 

I.I3579 

8.90686 

1.29143 

9 

i 

0-99357 

o.  11320 

0.99144 

0.13052 

0.98901 

o.  14780 

i 

2 

1.98714 

0.22640 

1.98288 

0.26105 

1  1.97803 

0.29561 

2 

3 

2.98071 

0.33960 

2-97433 

0.39157 

2.90704 

0.44342 

3 

4 

3.97428 

0.45281 

3.96577 

0.52210 

3.95606 

0.59123 

4 

30 

:  4.  96786 

o.  56601 

4.95722 

0.65263 

14.94508 

0.73904 

5 

30 

6 

5.96143 

0.67921 

5.94866 

0.78315 

:  5.93409 

0.88685 

6 

7 

6.95500 

0.79242 

6.94011 

0.91368 

6.92311 

1.03466 

7 

8 

7.94857 

0.90562 

7.93155 

1.04420 

7.91212 

1.18247 

8 

9 

8.94214 

1.01882 

8  .  92300 

I-I7473 

i  3.90114 

1.33028 

9 

i 

0.99306 

0.11753 

0.99086 

0.13485 

0.98836 

0.15212 

!  i 

2 

1.98613 

0.23507 

1.98173 

0.26970 

1.97672 

0.30424 

2 

2 

2.97920 

0.35261 

2.97259 

0.40455 

2.06508 

0.45637 

!  3 

4 

3.97227 

0.47014 

3-96346 

0.53940 

3.95344 

0.60849 

4 

45 

5 
6 

4-96534 

;  5.95841 

0.58768 

0.70522 

5.94519 

0.67425 
0.80910 

4.94180 
5.93016 

0.76061 

0.91274 

6 

ID 

7 

6.95147 

0.82276 

6.  9"  3606 

0-94395 

6.91853 

1  1.06486 

7 

8 

7.94454 

0.94029 

7.92692 

1.07880 

;  7.90689 

1.21698 

8 

9 

3.93761 

1.05783 

8.91779 

1.21365 

!  8.  89525 

1 

1.36911 

9 

5 

O 

En 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

g 

.  | 

P 

P 

.  c 

o 

Cfl 

3 
0 
fl 

:      8: 

1° 

S: 

1  '    8 

t° 

3 
0 

0 

P 

TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures — Continued. 


en 

o 

0 

9 

1C 

)° 

i 

t° 

,  o 

0 

in 

~? 

c 

C 

^ 

3 

rt 

re 

3 

C 
i 

to 

5 

Lat. 

Dep. 

Lat. 

Dep.  ' 

Lat. 

Dep. 

•  to 

|5 

C 

i 

0.98768 

0.15643 

0.98480 

0.17364 

0.98162 

0.19081 

j 

2 

1-97537 

0.31286 

1.96961 

0.34729 

1.96325 

0.38162 

1  .2 

3 

2  .  96306 

0.46930 

2.95442 

0.52094 

2.94488 

0.57243 

3 

4 

3-95075 

0.62573 

3.93923 

0.69459 

3.92650 

0.76324 

4 

0 

5 

4.93844 

0.78217 

4.92^03 

0.86824 

4-QQ8I3 

0.95405 

i  5 

60 

6 

5.92612 

0.93860 

5.90884 

1.04188 

5.88976 

1.14486 

!  6 

7 

6.91381 

1.09504 

6.89365 

I-2I553 

6.87130 

1.33566 

i  7 

8 

7.90150 

1-25147 

7-87846 

1.38918 

7-85301 

1.52648 

8 

9 

8.88919 

1.40791 

8.86327 

1.56283 

8.83464 

1.71729 

9 

i 

0.98699 

o.  16074 

0.98404 

0.17794 

0.98078 

0.19509 

i  i. 

2 

1-97399 

0.32148 

1.96808 

0.35588 

1.96157 

0.39018 

2 

3 

2  .  96098 

0.48222 

2.95212 

0.53383 

2.94235 

0.58527 

'  3 

4 

3.94798 

0.64297 

3-93616 

0.71177 

3.92314 

o.  78036 

1  4 

15 

5 

4.93498 

0.80371 

4.92020 

0.88971 

4.90392 

0-97545 

i  5 

45 

6 

5.92197 

0.96445 

!  5-90424 

1.06766 

5.88471 

1.17054 

i  6 

7 

6.90897 

1.12519 

;  6.  88828 

1.24560 

6.86549 

1.36563 

i  7 

8 

7.89597 

1.28594 

7.87232 

1.42354 

7.84628 

1.56072 

;  s 

9 

8.88296 

' 

i  .44668 

8.85636 

1.60149 

8.82706 

i.7558i 

9 

j 

i 

0.98628 

0.16504 

0.98325 

0.18223 

0.97992 

0.19936 

!  i 

2 

1.97257 

0.33009 

:  1.96650 

0.36447 

1.95984 

0.39873 

2 

3 

2.95885 

0.49514 

1  2.  94976 

0.54670 

2.93977 

0.59810 

3 

4 

3.94514 

0.66019 

3-  93301 

0.72894' 

3.91969 

0-79747 

4 

30 

5 

4.93142 

0.82523 

4.91627 

0.91117  1 

4.89962 

0.99683 

5 

30 

6 

5.9I77I 

0.99028 

;  5-  89952 

1.09341 

5.87954 

i  .  19620 

6 

7 

6.90399 

I-I5533 

6.88278 

1.27564; 

6.85947 

1-39557 

1  7 

8 

7.89028 

1.32038 

7.86603 

1.45788 

7.83939 

1-59494 

i  8 

9 

8.87657 

1.48542 

8.84929 

i  .64011 

8.81932 

1-79431 

i  Q 

i 

0.98555 

0.16935 

0.98245 

0.18652 

0.97904 

0.20364 

i 

2 

1.97111 

0.33870 

1.96490 

0.37304 

1.95809 

0.40728 

2 

3 

2.95666 

0.50805 

,  2.  94735 

0.55957 

2.93713 

0.61092 

1  3 

4 

3.94222 

0.67740 

3.92980 

0.74609 

3.91618 

0.81456 

4 

45 

5 

4.92778 

0.84675 

4-91225 

0.93262 

4-89522 

.01820 

!  5 

15 

6 

5.9*333 

i.  01610 

5.89470 

1.11914 

5.87427 

.22185 

i  6 

7 

6.89889 

1.18545 

6.87715 

1.30566 

6.8^331 

.42549 

;  7 

8 

7.88444 

1.35480 

7.85960 

1.49219 

7.83236 

.62913 

!  8 

9 

8.87000 

1.52415 

8.84205 

1.67871 

8.81140 

.83277 

9 

g 

¥ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0 

^ 

3 

3 

y 

!  P 

C 

c 
en 

3 
0 

o 

Sc 

>° 

7< 

| 

7* 

0 

3 
O 

0 

(/> 

TRAVERSE    TABLE. 


33 


Differences  of  Latitude  and  Departures — Continued. 


45 


12° 

' 

14° 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.    Dep. 

• 

jO.  97814 

0.20791 

•0.97437 

0.22495 

0.97029  0.24192 

1.95629 

0.41582 

1.94874 

0.44990 

1.94059 

0.48384 

2-93444 

0.62373 

1  2.923II 

0.67485 

2.91088 

0.72576 

3.91259 

0.83164 

1  3-  89748 

0.89980 

3.88118 

0.96768 

14.89073 

•03955 

.  4-87185 

1.12475 

4.85147 

1.20961 

5.86888 

.24747 

:  5.84622 

1.34970 

5.82177 

I.45I53 

6.84703 

.45538 

;  6.82059 

1.57465 

6.79206 

1-69345  : 

7.82518 

.66329 

7.79496 

i  .  70960 

7-76236 

1-93537 

8.80332 

.87120 

'8.76933 

2-02455 

8.73266  2.17729 

0.97723 

0.21217 

1  0.97337 

0.22920 

0.96923  !  0.24615 

1.95446 

0.42435 

1  1.94675 

0.45840 

1.93846  0.49230, 

2.93169 

0.63653 

2.92013 

0.68760 

2.9076910.73845 

13.90892 

0.84871 

3-89351 

0.91680 

3.87692  0.98461 

4.88615 

1.06088 

4.86689 

i  .  14600 

.4.84615  1.23076 

5.86338 

1.27306 

5-84027 

1-37520 

5.81538  1.47691 

6.84061 

1.48524 

6.81365 

i  .  60440 

6.78461  |  1.72307 

7.81784 

1.69742 

7.78703 

1.83360 

7.75384  1.96922 

,8.79507 

1.90959- 

8.  76041 

2.06280 

8.72307  2.21537 

0.97629 

0.21644 

;  0.97237 

0.23344 

0.96814  ;  0.25038 

1.95259 

0.43288 

I  1-94474 

0.46689 

1.93629  0.50076 

2.92888 

0.64932 

12.91711 

0.70033 

2.90444 

0.75114 

3.90518 

0.86576 

;  3.88948 

0.93378 

3.87259 

1.00152 

4.88148 

.08220 

,4.86185 

1.16722 

4.84073 

1.25190 

5.85777 

.29864 

15.83422 

i  .  40067 

5.80888 

1.50228 

6.83407 

.51508 

,  6.80659 

1.63411 

6.77703 

1.75266 

7.81036 

.73152 

17.77896 

1.86756 

7.74518 

2  .  00304 

8.78666 

.94796 

!8.  75133 

2.  IOIOO 

S.7I332 

2.25342 

0-97534  0.22069 
1.95068  0.44139 

0.97134  ,  o1.  23768 
1.94268  0.47537 

0.96704  0.25460 

1.93409  '0.50920 

2.92602  0.66209 

I  2.91402 

0.71305 

2.90113 

0.76380 

3.90136 

0.88278 

'3.88536 

0.95074 

3.86818 

I.OI84O  t 

4.87671 

.  10348 

14.85671 

1.18843 

4.83523 

I.2730I 

5-85205 

.32418 

i  5.82805 

I.426II  i 

5.80227 

1.52761 

6.82739 

.54488 

i  6.  79939 

1.66380 

6.76932 

I.7822I 

7-80273 

.76557 

7.77073 

1.90148  i 

7.73636 

2.03681 

8.77808 

.98627 

8.74207 

2.13917 

8.70341 

2.29141 

Dep. 

Lat. 

Dep. 

Lat. 

1 

Dep. 

Lat. 

77° 

76° 

75° 

60 


45 


34 


TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures  —  Continued. 

en 

o 

15°                             16' 

17° 

0 

, 

o 
c 

03 

- 

| 

0 

C 

s 

en 

Q 

Lat. 

Dep.     :      Lat. 

Dep. 

Lat. 

Dep. 

co 
Q 

c 

i 

0.96592 

0.25881    0.96126 

0.27563 

0.95630 

0.29237 

i 

2 

1.93185 

0.51763  ;  1.92252 

0.55127 

1.91260 

0.58474 

2 

. 

^ 

2.89777  [0.77645  ;;  2.88378 

0.82691 

2.86891 

0.87711 

3 

4 

3.86370  1.03527113.84504 

1.10254 

3.82521 

i  .  16948 

4 

O 

5 

4.82962 

1.29409    4.80630 

1.37818  i 

4-78152 

1.46185  |  5 

60 

6 

5.79^55 

1.55291  ;!  5.76757 

1.65382 

5.73782 

1.75423     6 

7 

6.76148 

I.8II73!   6.72883 

1.92946 

6.69413 

2.04660 

7 

a 

7.72740 

2.07055  i   7.69009 

2.20509 

7-65043 

2.33897 

8 

9 

8.69333 

2.32937     8.65135 

2.48073 

8.60674 

2.63134 

9 

1 

1 

i 

0.96478 

0.26303     0.96005 

0.27982 

0.95502 

0.29654 

i 

2 

1.92957 

O.  52606  ;    1  .92010 

0.55965 

,  1.91004 

0.59308 

2 

^ 

2.89436 

0.78909!   2.88015 

0.83948 

2.86506 

0.88962 

3 

15 

4 

5 

3.85914 
4.82393 

I.052I2 
I.3I5I5 

3.84020 
4.80025 

I.H93I 

1.39914 

3.82008 
4-77510 

i.  18616 
1.48270 

4 

5 

45 

6 

5.78872 

I.578I8 

5  •  76030 

1.67897 

5-73012 

1.77924 

6 

7 

6.75351 

I.S4I2I 

6.72035 

1.95880 

6.68514 

2.07579 

7 

8    ;7.7i82q 

2.10424     7.68040 

2.23863 

i  7.64016 

2.37233 

8 

9 

8.68308 

2.36728     8.64045 

2.51846 

8.59518 

2.66887 

9 

i 

0.96363 

0.26723 

0.95882 

0.28401 

0.95371 

0.30070 

i 

2 

1.92726 

0.53447      L9I764 

o.  56803 

1.90743 

0.60141 

2 

2.89089 

0.80I7I      2.87646 

0.85204 

2.86115 

o  .  902.1  1 

3 

4 

3.85452 

1.06895    :  3.8352S 

1.13606 

!  3.81486 

1.20282 

4 

30 

5 

4.81815 

1.33619 

4.79410 

1.42007 

14.76858 

1.50352 

5 

30 

6 

5.78178 

1.60343!',  5.75292 

i  .  70409 

1  5.72230 

1.80423 

6 

7 

6.74541 

1.87066  i  6.71174 

1.98810 

6.67601 

2.10494 

7 

8 

7  .  70Q04 

2.13790  ;  7.67056 

2.27212 

7.62973 

2.40564 

8 

9    18.67267 

2.40514  |  8.62938 

2.55613 

8.58345 

2.70635 

9 

|| 

1 

i 

0.96245 

0.27144 

0-95757 

0.28819 

0.95239 

0.30486 

i 

2 

1.92491 

0.54288 

1.91514 

0.57639 

1.90479 

0.60972 

o 

o 

2.88736 

0.81432 

2.87271 

0.86458 

'  2.85718 

0.91459 

3 

4 

3.84982 

1.08576 

3.83028 

1.15278 

1  3.80958 

1.21945 

4 

45 

5 

i  4.81227 

1.35720 

14.78785 

i  .  44098 

i!  4.76197 

1.52432 

|  5 

15 

6 

5-77473 

1.62864 

15.74542 

1.72917 

j!  5-  71437 

1.82918 

b 

7 

6.73718 

1^90008   6.70299 

2.01737 

!;6.66677 

2.13405 

7 

8 

7.69964 

2.17152 

7-66057 

2.30557 

!  7.61916 

2.43891 

8 

9 

8.66209 

2  .  44296 

I  8.61814 

2.59376 

\  8.57156 

2.74377 

9 

:K 

srll   DeP- 

Lat. 

;     Dep. 

Lat. 

Dep. 

Lat. 

g 

En" 

5 

3 

H 

P 

3 

1 

3 

(5 
O 

73°                         .    72° 

C/l 

o 
o 

74° 

CD 

TRAVERSE    TABLE. 


35 


Differences  of  Latitude  and  Departures — Continued. 


c/2 

0) 

18° 

19° 

20° 

1  c 

!  ^ 

.8 

rt 

VI 

3 

|5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 

c 

j 

0.95105 

0.30901 

0.94551 

0.32556 

0.93969  !  0.34202 

i 

2 

i.  9021  i 

0.61803 

1.89103 

0.65113 

1.87938  10.68404 

2 

3 

2.85316 

0.92705 

2.83655 

0.97670  2.81907  1.02606 

•3 

i  4 

3.80422 

1.23606  i;  3.78207 

1.30227 

3.75877  1-36808 

4 

0 

i  5 

4.75528 

1.54508 

4.72759 

1.62784 

!  4.69846  j  1.71010 

60 

b 

5  •  70633 

1.85410  5-67311 

1.95340 

5.63815  2.05212 

6 

!  7 

6.65739 

2.  16311   6.61863 

2.27897 

6.57784 

2.39414 

7 

!  8 

7.60845 

2.47213 

7.56414 

2.60454 

7.51754 

2.73616 

8 

'  9  8.55950 

i   I 

2.7SII5 

8.50966 

2.93011 

8.45723 

3.07818 

9 

i 

0.94969 

0.31316 

0.94408 

0.32969 

0.93819 

0.34611 

i 

2 

1.89939  0.62632  ::  I.S83I7 

0.65938 

1.87638:0.69223 

2 

.3 

2.84909  0.93949   2.83226 

0.98907 

2.81457  1.03835 

3 

4 

3.79879  L25265  ||  3.77635 

1.31876 

3.75276  1.38446 

4 

15 

5 

4.74849 

1.56581   4.72044 

1.64845 

4.69095  1.73058 

45 

6 

5.69819 

1.87898   5-66453 

1.97814  5.62914  2.07670 

6 

7 

6.64789 

2.19214   6.60862 

2.30783 

6.56733J2.4428I 

7 

8 

7-59759 

2.50531 

7.55271 

2.63752 

7.50553  2.76893 

8 

9  8.54729 

2.81847  |  8.49680 

2.96721 

8.44372 

3-11505 

9 

| 

i 

0.94832 

0.  31730  :!  0.94264 

0.33380 

O.93667  !  O.35O2O 

i 

2 

1.89664 

0.63460::  1.88528 

0.66761  1.87334  0.70041 

2 

3 

2.84497 

0.95191  j  2.82792 

1.00142 

2.8IOOI 

1.05062 

3 

4 

3.79329 

I.2692I  ! 

3.7/056 

1.33522 

3.74668 

1.40082 

4 

30 

i 

4.74161 
5-68994 

1.58652   4.71320 
1.903821  5.65584 

1.66903 
2.00284 

4.68336 
5.62003 

1.75103 
2.10124 

5 
6 

30 

7 

6.63826 

2.22113  !  6.59849 

2.33664 

6.55670 

2.45145 

7 

8 

7-58658 

2.53843!  7.54H3 

2.67045 

7.49337 

2.80165 

8 

9 

8-53491 

2.85574  8.48377 

3.00426 

8  .  43004 

3.15186 

9 

i 

0.94693 

0.32143  0.94117 

0.33791 

0.93513 

0.35429 

i 

2 

1.89386 

0.64287,  1.88235 

0.67583 

1.87027  0.70858 

2 

3 

2.84079 

0.96431  2.82352 

LOI375 

2.80540  1.06287 

2 

4 

3.78772 

1-28575  3.76470 

1.35166 

3.74054  1.41716 

4 

45 

5 

4.73465 

1.60719 

4-70588 

1.68958 

4-67567  I.77I4* 

5 

15 

b 

5.68158 

1.92863  ; 

5.64705 

2.02750 

5.6loSl  j  2.  12574 

6 

7 

6.62851 

2.25007 

6.58823 

2.36541 

6.54594  2.48003 

7 

y 

7-57544 

2.57151  1 

7.52940 

2.70333 

7.48108  2.83432 

8 

9 

8.52237 

2.89295  j 

8.4/058 

3.04125 

8.41621 

3.18861 

9 

i 

3' 

s 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0 

§ 

c 

53 

P 

c 

.w 

71° 

70° 

69° 

0 
(0 

o 

M 

36                                                     TRAVERSE    TABLE. 

Differences  of  Latitude  and  Departures  —  Continued. 

. 

. 

21° 

22° 

23° 

. 

C/J 

O 

O 

! 

o 

c 

0 

G 

i 

rt 

CO 

5 

Lat. 

Dep.    j 

Lat. 

Dep. 

Lat. 

Dep. 

rt 
tfl 

5 

3 
C 

s 

I  r 

0.93358 

0.35836  |j 

0.92718 

0.37460  !  0.92050 

0.39073 

, 

2 

.86716 

0.71673  ! 

1.85436 

O.7492I     ;  I.84IOO 

0.78146 

2 

'  3 

.  80074 

1.07510  i 

2.78155 

I.I238I 

2.76151 

1.17219 

3 

4 

.73432 

1-43347 

3-70873 

1.49842 

3.6S20f 

1.56292 

4 

O 

5 

4.66790 

1.79183 

4.63591 

I.S7303       4.60252 

1.95365 

5 

60 

6 

.60148 

2.  I502O 

5.56310 

2.24763    i  5.52302 

2.34438 

b 

7 

6.53506 

2.50857 

6.49028 

2.62224       6.44353 

2.735H 

7 

8 

7.46864 

2.86694; 

7.41747 

2.99685       7.36403 

3.12584 

8 

9 

8.40222 

3.22531 

8.34465 

3.37145       8.28454 

3-51657 

9 

T 

0.93200 

0.36243 

0.92554 

0.3/864 

0.91879 

0.39474 

i 

2 

1.86401 

0.72487 

1.85108 

0.75729    j  1.83758 

0.78948 

2 

0 

2  .  79602 

I.OS73I 

2.77662 

I.I3594    :   2.75637 

i.  -i  8423 

3 

4 

3.72803 

1-44975 

3.70216 

I.51459    3.67516 

1.57897 

4 

15 

=1 

4  .  66004 

1.81219  i 

4.62770 

1.89324  14-59395 

1.97372 

5 

45 

6 

5.59204 

2.17462  ! 

5'.  55324 

2.27189 

5.51274 

2.36846 

b 

7 

6.52405 

2.53706 

6.47878 

2.65054    6.43153 

2.76320 

7 

8 

7.45606 

2.89950 

7-40432 

3.02918 

7.35032 

3-15795 

8 

9 

8.38807 

3.26194 

8.32986 

3.40783    8.26912 

3-55269 

9 

i 

0.93041 

0.36650 

0.92388 

0.38268    0.91706 

0.39874 

i 

2 

1.86083 

0.73300  ' 

1.84776 

0.76536    1.83412 

0.79749 

2 

2.79125 

1.09950 

2.77164 

1.14805    2.75118 

1.19624 

O 

4 

3.72167 

1.46600 

I3.69552 

1.53073 

3.66824 

1-59499 

i  4 

30 

5 
6 

4.65208 
5.58250 

1.83250 

4-61940 

I.9I34I 
2.29610 

5-50236 

1-99374 
2.39249 

6 

30 

7 

6.51292 

2.56550 

!6.  46716 

2.67878 

6.41942 

2.79124 

7 

8 

7.44334 

2.93200 

7.39104 

3.06146 

7.33648 

3.18999 

8 

9 

8-37375 

3.29851 

8.31492 

3.44415 

8.25354 

3-58874 

9 

i 

0.92881 

0.3/055 

0.92220 

0.38671 

0.91531 

0.40274 

i 

2 

1.85762 

0.74111 

1.84440 

0.77342  |  1.83062 

0.80549 

2 

'12.78643. 

1.11167 

2  .  76660 

I.I60I3    :  2.74593 

1.20824 

,  3 

4 

3.7I524 

1.48222 

3.68880 

1.54684 

3.66124 

1.61098 

4 

45 

5 
6 

4.64405 
5.57286 

1.85278 
2.22334 

4.6IIOO 

5-53320 

1-93355 
2.32O26 

4-57^55 
5.49186 

2.01373 
2.41648 

5 
6 

15 

7 

6.50167 

2.59390 

6.45540 

2.70697 

6.40718 

2.81922 

7 

8 

7.43048 

2.96445 

:  7.  37760 

3.09368 

7.32249 

3-22197 

8 

9 

;s.  35929 

3;  33501 

8.29980 

3.48039 

8.23780 

3-62472 

,  9 

5 

5' 

(3 

O 
on 

p* 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

g 

5 

rt 

t/i 

n 

ft) 

68° 

i              67° 

66° 

n 
o 

o 

'73 

TRAVERSE    TABLE. 


37 


Differences  of  Latitude  and  Departures — Continued. 


45 


24° 

25° 

26° 

0 

w 

B 

rt 

"2 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

to 

5 

C 

§ 

0.91354 

0.40673 

0.90630 

0.42261 

0.89879 

0.43837 

i 

1.82709 

0.81347 

1.81261 

0.84523 

1.79758 

0.87674 

i  2 

2.74063 

I.22O2O 

2.71892 

1.26785 

2.69638 

1.31511  3 

3.65418 

.1.62694 

3.62523 

1.69047 

3.59517 

1.75348  :  4 

4.56772 

2.03368 

4.53153 

2.11309 

4-49397 

2.19185  :  5 

60 

5-48127 

2.44041 

5.43784 

2.53570 

5.39276 

2.63022  \  6 

6.39481 

2.84715 

6.344T5 

2.95832 

6.29155 

3.06859  7 

7.30836 

3.25389 

7.25046 

3-38094 

7.19035 

3.50696 

!  8 

8.22190 

3  .  66062 

8.15677 

3.80356 

8.08914 

3-94533 

!  Q 

0.91176 

0.4^071 

0.90445 

0.42656 

0.89687 

0.44228 

j 

1.82352 

0.82143 

1.80891 

0.85313 

1-79374 

0.88457 

:  2 

2.73528 

1.23215 

2.71336 

1.27970 

2.69061 

1.32686 

3 

3-64704 

1.64287 

3.61782 

1.70627 

3.58749  1.76915  4 

4.55881 

2.05359  4.52227  2.13284 

4.48436  2.21144  5 

45 

5.47057 

2.46431  !!  5.42673  2.55941 

5.38123  2.65373  6 

6.38233 

2.87503  1  6.33118  2.98598 

6.27810  3.09602  ':  7 

7.29409 

3.28575  7-23564  3-41254 

7.17498  3-53830:  8 

8.20585 

3.69647  18.14009  '3.  83911 

8.07185  3.98059  9 

i 

I. 

0.90996 

0.414691:0.90258  0.43051 

0.89493 

0.44619 

i 

1.81992 

0.82938  :  i  .80517  0.86102 

1.78986 

0.89239   2 

2.72988  1.24407  2.70775 

1.29153 

2.68480 

I.33859 

i  3 

3.63984  1.65877  3.61034 

1.72204 

3.57973 

1.78479 

1  4 

4.54980  2.07346  4.51292 

2.15255 

4.47467 

2.23098 

,  5 

30 

5.45976 

2.48815  5.4155112.58306 

5.36960 

2.67718 

j  6 

6.36972 

2.90285 

6.31809  3-01357 

6.26454  3-12338 

7 

7.27969 

3-31754 

7.22068  3.44408 

7.15947  3.56958  ;  8 

8.18965 

3.73223 

8.12326  3.87459 

8.05440 

4.01578  9 

0.90814 

0.41866  0.90069  0.43444 

0.89297 

0.45009 

i 

1.81628 

0.83732  1.80139 

0.86889 

1.78595  O.gOOig   2 

;  2.  72442 

1.25598 

2.70209  1.30333 

2.67893  1.35029  3 

'3.63257 

1.67464  j  3.60279  1.73778 

3.57191  1.80039  4 

4.54071 

2.09330  4-50349  2.17222 

4.46489  2.25049!  5 

15 

S5-44885 

2.51196 

5.40418  2.60667 

5.35787 

.2.70059  6 

6.35700 

2.93062 

6.30488  3-04111 

6.25085 

3.  15068  1  7 

7.26514 

3-34928 

7.20558  3.47556 

7.14383 

3.60078  8 

8.17328  3.76794 

8.10628  3.91000 

8.03681 

4.05088  9 

i 

j 

Dep.    Lat. 

Dep. 

Lat. 

Dcp. 

Lat. 

on' 

§ 
5" 

'         '          I 

P 

£ 

65°                64° 

63° 

P 

en 

3§                                                     TRAVERSE  TABLE. 

Differences  of  Latitude  and  Departures  —  Continued. 

i/i 

22 

o 
o 

c 

27° 
1 

28°                           29° 

o 

3 

in 

o 

3 

ri 

rJ 

3 

c 

i 

Q 

Lat. 

Dep.          Lat. 

Dep.           Lat. 

Dep. 

"t/2 

Q 

3 

i 

0.89100   0.45399 

0.88294 

0.46947  !  0.87462 

0.48481 

i 

2 

1,78201 

0.90798 

1.76589   0.93894    1.74924 

0.96962 

2 

3 

2.67301 

1.36197 

2.648841  1.40841    2.62386 

1-45443 

3 

4 

3-56402 

1.81596 

3.53179    1.87788:  3-49848 

1.93924 

4 

o 

5 

4.45503 

2.26995 

4.41473 

2-34735  :  4.37310 

2.42405 

5 

60 

6 

5.34603 

2.72394 

5-29768 

2.81682    5.24772 

2.90886 

6 

7 

6.23704 

3-17793 

6.18063 

3.28630    6.12234    3.39367 

7 

8 

7.12805 

3.63193 

7.06358 

3-75577  1:6.99696 

3-87848 

8 

9 

8.01905 

4-08591 

7.94652 

4.22524    7-87156 

4-36329 

9 

i 

1          '        r 

i 

0.88901    0.45787  1 

0.88089:0.47332    0.87249 

0.48862 

i 

2 

1.7780310.915741  1.76178  (0.94664     1.74499 

0.97724 

2 

3 

2.66705    1.37362     2.64267    1.41996    2.61748 

1.46566 

3 

4 

3.556o6 

1.83149 

3.52356 

1.89328    3.48998 

r.95448 

4 

15 

5 

4.44508 

2.28937 

4.40445 

2.36660;  4.36248 

2.44310 

5 

45 

6 

5.33410 

2.74724 

5-28534 

2.83992    5-23497 

2.93172 

6 

7 

6.22311 

3.20511 

6.  16623 

3.31324    6.10747 

3.42034 

7 

8 

7.11213    3.66299 

7.04712 

3.78656    6.97996 

3.90896 

8 

9 

8.00115    4-  12086     7.92801    4.25988,  7.85246 

4-39759 

9 

i   :  0.88701 

0.46174 

0.87881 

0.47715    0.87035 

0.49242 

i 

2 

1.77402    0.92349 

1.75763 

0.95431  |:  1.74071 

0.98484 

2 

3 

2.66103  !  1.38524 

2.63645 

I.43I47  j,  2.  6IIO6 

1.47727 

3 

4 

3.54804    1.84699 

3.5I526 

1.90863  :  3.  48142 

i  .  96969 

4 

. 

30 

5 

4.43505 

2.30874 

4o94o8 

2.38579!  4.351/7 

2.46211 

=; 

30 

6 

5.32206 

2.77049 

5.27290    2.86295  -:  5.22213 

2-95454 

6 

7 

6.20907 

3.23224 

6.15171    3.34011     6.09248 

3.44696 

7 

8 
9 

7.09608 
7-98309 

3-69398; 
4.15573 

7-03053 
7.90935 

3.81727  :    6.96284 
4.29442       7.83320 

3.93938 
4.43i8i 

8 
9 

i 

;  0.88498 

0.46561 

0.87672 

0.48098      O.S68I9 

0.49621 

i 

2 

'1.76997 

0.93122 

1-75345 

0.96197!'  1.73639 

•0.99243 

2 

3 

2.65496 

i  .  39684 

2.63018 

1.44296  i    2.60459 

I  1.48864 

3 

4 

3-53995 

1.86245 

3.50690 

1.92395  ,'  3.47279 

1.98486 

4 

45 

5 

4.42493 

2.32807 

408363 

2.40494!    4.34099 

|  2.48108 

5 

15 

6 

5-30992 

2.79368 

5  .  26036 

2.885931    5-20919 

2.97729 

6 

7 

!  6.  19491 

3.25930 

6.13708 

3.36692,!6.07739 

3-47351 

7 

8    i  7-07990 

3.72491     7.01381 

3.84791;    6.94559 

3.96973 

8 

4-  19053     7-89054 

4.32889      7.81378 

| 

4.46594 

9 

3' 

O 

en 

Dep. 

Lat.          Dep. 

Lat.          Dep. 

Lat. 

2 

en' 

g 

3' 

p* 

p 

3 

»—  * 

3 

3 

r? 

;/) 

0 

n> 

62° 

61°                              60° 

P 

00 

TRAVERSE    TABLE. 


39 


Differences  of  Latitude  and  Departures — Continued. 


irt 

U 

<u 
u 

53 

30° 

31° 

32° 

o 
o 

in 

<L> 

3 

rt 

i 

rt 

3 

C 

s 

tfl 

Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o> 

5 

I 

0.86602 

0.50000 

0.85716 

0.51503 

0.84804 

0.52991 

i 

2 

'1.73205 

I  .  00000 

I.7I433 

•  03007 

T  .  69609 

1.05983 

2 

3 

2.59807 

1.50000  2.57150 

•545II 

2.54414 

L5S975 

3 

4 

3.46410 

2.OOOOO   3.42866 

.06015 

3.39219 

2.  11967 

4 

0 

5 

4-33012 

2.50000 

4-28583 

.57519 

4.24024 

2.64959 

5 

60 

6 

5-19615 

3  .  ooooo 

5.14300 

.09022 

5.08828 

3.I795I 

6 

7 

6.06217 

3  .  50000 

6.00017 

.60526 

5.93633 

3.70943 

7 

8 

6.92820 

4.00000  6.85733 

4.12030 

6.78438 

4.23935 

8 

9 

7.79422 

4.50000!  7.71450 

4.63534 

7.63243 

4.76927 

9 

i 

0.86383  ; 

0.50377 

0.85491 

0.51877 

0.84572 

0.53361 

i 

2 

1  1.72767  i 

1.00754!!  1.70982 

1.03754 

1.69145 

1.06722   2 

3 

12.59150 

1.51132 

2.56473 

1.55631 

2.537IS 

I  .60084 

3 

4 

3-45534 

2.01509 

3.41964 

2.07509 

3.38291 

2.13445 

4 

T5   5 

2.51887  4.27456 

2.59386 

4.22863 

2.66807 

5 

45 

6 

5.18301 

3.02264 

5-12947 

3.11263 

5.07436 

3.2OI68 

6 

7 

6.04684 

3.52641  5.98438 

3-63141 

5  -  92009 

3.73530; 

7 

8 

6.91068 

4.03019  1  6.83929 

4.  15018 

6.76582 

4.26891 

8 

9 

7-77451 

4.53396  :  7.69420 

4.66895 

7-6ii55 

4.80253 

9 

j 

;  0.86162 

0.50753  1  0.85264 

0.52249 

0.84339 

0-53730 

i 

2   1.72325 

1.01507  1.70528 

1.04499 

1.68678 

I  .07460 

2 

3 

2.58488 

1.52261  j  2.55792 

1.56749 

2.53017 

I.  6ligO 

3 

4 

3.44651 

2.03015 

3.41056 

2.08999 

3-37356 

2.  14920 

4  ! 

30   5 

4-30814 

2.53769,  4.  26320 

2.61249 

4-21695 

2.68650 

5 

30 

i  6 

5-16977 

3.04523  5.11584 

3.13499 

5.06034 

3.22380 

6 

7 

6.03140 

3-55276  !  5-96948 

3.65749 

5.90373 

3.76110 

7 

8 

6.89303 

4.06030  |  6.82112 

4.17998 

6.74713 

4.29840 

8 

9 

7.75466 

4.56784  7-67376 

4.70248 

7-59052 

4-83570 

9 

i 

0.85940 

0.51129  0.85035 

0.52621 

0.84103 

0.54097 

i 

2   I.7I88I 

1.02258  1.70070 

1.05242 

!  1.68207 

1.08194 

2 

3 

2.57821 

1.53387  2.55105 

1.57864 

2.52311 

I  .62292 

3 

1  4  I3.43762 

2.04517 

3.40140 

2.10485 

3-36415 

2.16389 

4 

45  !  5  4.29703 

2.55646 

4.25176 

2.63107 

4.20519 

2.70487 

5   15 

6  5.15643 

3.06775 

5.10211 

3.15728 

5.04623 

3.24584 

6 

7 

6.01584 

3.57905 

5.95246 

3-68349 

5.88827 

3.78682   7 

S  6.87525(4.09034 

6.80281 

4.20971 

6.72831 

4-32779  8 

9  7.73465 

4.60163  7.65316 

4.73592 

7-56935 

4.86877  9 

>" 

£3 

2  1  Dep. 

(/) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.    2 

e   as 

P 

c 

e? 

vi 

1       59° 

58° 

57°      \  8 

0 
Cfl 

TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures  —  Continued. 


15 


30 


45 


<u 

33° 

34° 

35° 

oJ 

u 

o 

c 

e<j 

rt 

to 

5 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

tf3 
Q 

T 

0.83867 

0.54463 

0.82903  0.559-19 

0.81915 

0-57357 

I 

2 

1.67734 

1.08927 

1.65807  1.11838 

1.63830 

I.I47I5 

2 

3 

2.51601 

1-63391 

2.48711  1.67757 

2-45745 

1.72072 

3 

4 

3.35468 

2.17055 

3.31615 

2.23677 

3.27660 

2.29430 

4 

5 

4.19335 

2.72319 

4.I45I8 

2.79596 

4.09576 

2.86788 

5 

6 

5.03202 

3-26783 

'4-97422 

3.35515 

4.91491 

3.44I45 

6 

7 

5.8706913.81247 

5.80326  3.91435 

5.73406 

4.01503 

7 

8 

6.70936 

4-357II 

6.63230  4.47354 

6.55321 

4.58861 

8 

9 

7.54803 

4.90175 

7.46133  5-03273 

7.37236 

5.16218 

9 

i 

0.83628 

0.54829 

0.82659  0.56280 

0.81664 

0.57714 

i 

2 

1.67257 

1.09658 

1.65318 

1.12560 

1.63328 

1.15429 

2 

3 

2.50885 

1.64487 

2.47977 

1.68841 

2.44992 

I.73I43 

3 

4 

3.34514 

2.19317 

3.30636 

2.25121 

3.26656 

2.30858 

4 

5 

4.18143 

2.74146 

4.13295 

2.81402 

4.08320 

2.88572 

5 

6 

5-01771 

3-28975 

4.95954 

3.37682 

4.89984 

3-46287 

6 

7 

5.85400 

3-83805 

5-78613 

3-93963 

5.71649 

4.04001 

7 

8 

6.69028 

4.38634 

6.61272 

4-50243 

6.53313 

4.61716 

8 

9 

7-52657 

4.93463 

7-43931 

5.06524 

7-34977 

5  .  19430 

9 

i 

0.83388 

0.55193 

0.82412 

0.56640 

0.81411 

0.58070 

i 

2 

1.66777 

1.10387 

1.64825 

1.13281 

1.62823 

i.  16140 

2 

3 

2.50165 

1.65581 

:2.  47237 

1.69921 

2.44234 

1.74210 

3 

4 

3-33554 

2.20774 

3.29650 

2.26562 

3.25646 

2.32281 

4 

5 

4.16942 

2.75968 

4.  I2O63 

2.83203 

4-07057 

2.90351 

5 

6 

5.00331  3.31162 

14-94475 

3.39843 

4.88469 

3.48421 

6 

7 

5.83720  3.86355 

5.76888 

3.96484 

5.69880 

4.06492 

7 

8 

6.67108  4.41549 

6.59300 

4.53124 

6.51292 

4.6-f562 

8 

9 

7.50497  4.96743 

7-4I7I3 

5.09765 

7-32703 

5.22632 

9 

| 

i 

0.83147  '0.55557 

0.82164 

0.56099 

0.81157  ,0.58425 

i 

2 

1.66294  1.  1  1114 

1.64329 

1.13999 

1.62314 

1.16850 

2 

3 

2.49441  1.66671 

2.46494 

i  .  70999 

2.43472 

1.75275 

3 

4 

3.32588 

2.22228 

3.28658 

2.27998 

3.246291  2.33700 

4 

5 

4-15735 

2.77785 

4.10823 

2.84998 

4.05787 

2.92125 

6 

4.98882 

3.33342 

14.92988 

3.41998 

4.86944 

3-50550 

6 

7 

5.82029 

3.88899 

5.75152 

3.98997 

5.68101 

4.08975 

7 

8 

6.65176 

4-44456 

6.57317 

4-55997 

6.49260 

4.67400 

8 

9 

7.48323  5.00013 

7.39482 

5.12997 

7.30416  5.25825 

9 

|   Dep.    Lat. 

Dep. 

Lat. 

Dep.    Lat. 

D 

E' 

5? 

? 

3 

3 

3 

56' 

55° 

54° 

0 

o 

60 


45 


30 


15 


o  i  g 


TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures — Continued.. 


M 

OJ 

a) 
o 

36° 

37° 

38° 

§ 

t« 

0 

3 

-2 

2 

3 

a 

3 

tfl  ! 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

to 

Q 

c 

i 

j 

0.80901 

0.58778 

0.79863 

o.  60181 

0.78801 

0.61566 

i 

2 

1.61803 

I.I7557 

1.59727 

1.20363 

1.57602 

1.23132 

2 

3 

2.42705 

I.76335 

2.39590 

1.80544 

2  .  36403 

1.84698 

3 

4 

3.23606 

2.35H4 

3-19454 

2.40726  3.15204 

2.46264  4 

0 

5 

4.04508 

2.93892 

3.99317 

3-00907 

3.94005 

3.07830  5 

60 

6 

4.85410 

3.52671 

4.79181 

3.61089  4.72806 

3-69396 

6 

7 

5-66311 

4.11449 

5.59044 

4.21270  j  5.51607 

4.30963 

7 

8 

6.47213 

4.70228 

6.38908 

4.81452  6.30408  14.92529 

8 

9 

7.28115 

5  .  29006 

7.18771 

5.41633  7.09209 

5.54095 

9 

i 

0.80644 

0.59130 

0.79600 

0.60529  1.78531 

0.61909 

i 

2 

1.61288 

1.18261 

1.59200 

1.21058  |  1.57063 

1.23818 

2 

3 

2.41933 

1.77392 

2.38800 

1.81588  2.35595 

1.85728 

3 

4 

3-22577 

2.36523 

3.18400 

2.42117 

3.14126 

2.47637 

4 

15 

5 

4.03222 

2.95654 

3.98001 

3-02647 

3.92658 

3.09547 

5 

45 

6 

4.83866 

3.  54785 

4.77601 

3.63176 

4.7H90 

3.7T456 

6 

7 

5.645H 

4.13916 

5-57201 

4-23705!'  5-49721 

4.33365 

7 

8 

6.45155 

4.73047 

6.36801 

4.84235  6.28253 

4.95275 

8 

9 

7.25800 

5-32178 

1  7.16401 

5.44764 

7.C6785 

5.57184 

9 

I 

i 

0.80385  :  0.59482 

0-79335 

0.60876 

0.78260 

0.62251 

i 

2 

I.6077I   1.18964 

1.58670 

1.21752 

I.5652I 

1.24502 

2 

3 

2.4II57   1.78446 

2.38005 

1.82628  2.34782 

1.86754 

3 

4 

3.21542   2.37929 

3.  17341 

2.43504  I3.I3043 

2.49005 

4 

30  5 

4.01928   2.974II 

13.96676 

3.043801  3-  9*304 

3.H257 

5 

30 

6 

4.82314  3.56893 

4.76011 

3.65256  4.69564 

3.73508 

6 

7 

5.62699 

4.16375 

iS-55347 

4.26132  ;  5.47825 

4.3576o 

7 

8 

6.43085   4.75858 

6.34682 

4.87OO9   6.26O86 

4.98011 

O 

9 

7.23471   5.35340 

7.14017 

5.47S85    7-04347 

5.60263 

9 

i 

O.8OI25  0.59832 

0.79068 

0.61221  O.-7798S 

0.62592 

i 

2 

I.6O25O   1.19664 

1-58137 

1.22443  L55946 

1.25184 

2 

3 

2.40376  1.79497 

(2.37206 

1.83665 

2.33965 

1.87777 

3 

4 

3.20501  12.39329 

13.16275 

12.44886 

3-IT953 

2.50369 

4 

45 

5 

4.00626 

2.99162 

3-95344 

I  3.06108 

3  •  89942 

3.12961   5 

15 

6 

4-80752 

3.58994 

4.74413 

3.67330 

4.67930 

3-75554  ,  6 

7 

5-60877 

4.18827 

15.53482 

4.28552 

5.45919 

4.38146  i  7 

8 

6.41003 

4.78659 

6.32551 

4.89773 

6.23907 

5.00738   8 

9 

7.21128  5.38492 

7.  11620 

;  5.50995 

7.01896 

5.63331 

;  9 

§ 

g 

VI 

Dep.    Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

g 

g 

c 

i 

£ 

e 

of 

CO 

p 

0 
CD 

53° 

52° 

p 

51°     I  8 

o 
tn 

42                                                     TRAVERSE    TABLE. 

Differences  of  Latitude  and  Departures  —  Continued. 

t/3 

4) 

39° 

40° 

4i° 

d 

t/5 

o 

~J 

o 

3 

g 

c 
rt 

3 

a 

W 

C 

S 

Q 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Q 

s 

I 

0.77714 

0.62932 

0.76604 

0.64278 

0.75470 

0.65605 

i 

2 

1-55^29 

1.25864 

i.  53208 

1.28557 

1.50941 

1.31211 

2 

3 

2.33143 

1.88796 

2.29813 

1.92836 

2.26412 

1.96817 

3 

4 

3.10858 

2.51728 

3.06417 

2.57H5 

3.01883 

2.62423 

4 

o 

5 

3.38573 

3.14660 

3-83022 

3-21393 

i3-77354 

3.28029 

5 

60 

6 

4.66287 

3.77592 

4.59626 

3.85672 

4.52825 

3:93635 

6 

X" 

7 

5.44002 

4.40524 

5-36231 

4-49951 

5.28296 

4.59241    ;    7 

8 

6.21716 

5-03456 

6.12835 

5.14230 

6.03767 

5.24847     8 

9 

6.99431 

5.66388 

6.89439 

5.78508 

6.79238 

5.90453     9 

i 

0-77439 

0.63270 

0.76323 

0.64612 

0.75184 

0.65934 

i 

2 

1.54878 

1.26541 

1.52646 

1.29224 

i.  50368 

1.31869  ;  2 

3 

2.32317 

1.89811 

2.28969 

I.93S37 

2.25552 

1.97803 

3 

4 

3-09757 

2.53082 

3-05293 

2.58449 

3.00736 

2.63738 

4 

15 

5 

3.87196 

3-16352 

3.81616 

3.23062 

3.75920 

3.29672 

5 

45 

6 

4.64635 

3.79623 

4-57939 

3.87674 

.4.51104 

3-95607 

6 

7 

5.42074 

4.42893 

5.34262 

4.52286    5.26288 

4-61542 

7 

8 

6.19514 

5.06164 

6.10586 

5.  16899  ||  6.01472 

5.27476 

8 

Q 

6.96953 

5.69434 

6.86909 

5.8I5II 

6.76656 

5-934II 

9 

r 

0.77162 

0.63607 

0.76040 

0.64944        0.74895 

0.66262 

i 

3 

1.54324 

2.31487 

1.27215 
1.90823 

1,52081 
2.28121 

1.29889     I.49791 
1.94834    2.24686 

1.32524       2 

1.98786     3 

4 

3-08649 

2.54431 

3.04162 

2-59779    2.99582 

2.65048 

4 

30 

5 

3.85812 

3.18039 

3.80203 

3.24724    3-74477 

3-31310 

5 

30 

6 

4.62974 

3.81646 

4-56243 

3.89668 

4-49373 

3-97572 

6 

7 

5.40137 

4.45254 

5.32284 

4.54613 

5.24268 

4-63834 

!   7 

8 

6.17299 

5.08862 

6.08324 

5.19558115-99164 

5  -  30096 

,  8 

9 

6.94462 

5.72470 

6.84365 

5.84503    6.74060 

5-96358 

'  9 

i 

0.76884 

0.63943 

0.75756 

0.65276  !  0.74605 

0.66588 

i 

2 

1.53768 

1.27887 

L5I5I3 

1.30552     ;    I.492II 

I.33I76 

2 

3 

2.30652 

1.91831 

2.27269 

1.95828     |    2.23817 

1.99764 

1  3 

4 

3.07536 

2-55775 

3.03026 

2.6II04 

2.98422 

2.66352 

4 

45 

5 

3  .  84420 

3.I97I9 

3-78782 

2.26380        3.73028 

3-32940 

'  5 

15 

6 

4-61305 

3.83663 

4-54539 

3.91656     \   4.  47634 

3.99529 

6 

7 

5.38189 

4.47607 

5-30295 

4.56932     15.22240 

4.66117 

1  7 

8 

6.15073 

5.H55I 

6.06052 

5.22208         5.96845 

5-32705 

•  8 

9 

6.91957 

5-75495 

6.81808 

5.87484 

6.71451 

5.99293 

;  9 

3 

3' 

O 

Cfl' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0 

GO 

g 

5 

B 

p 

!     P 

c 

(t 

on 

0 

n> 

50° 

49° 

48  ' 

;   o 
a> 

0 

en 

TRAVERSE    TABLE. 


43 


Differences  of  Latitude  and  Departures  —  Continued. 

C/3 

o 

42° 

43° 

44°  .            ;;  o 

in 

CJ 

£ 

'        C 

0 

a 

rt 

3 

a 

i 

tfl 

5  : 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

OQ 

Q 

1 

1 

jl 

i  •  o  74314 

0.66913 

0.73135 

0.68199    0.71933 

0.69465; 

I 

2       1.48628 

1.33826 

1.46270 

1.36399!  1.43867 

1-38931 

2 

fj 

2.22943 

2.00739 

2.  19406 

2.04599  i  2.15801 

2.08397 

3 

4 

2.97257 

2.67652 

2.92541 

2.72799    2.87735 

2.77863,    4 

0 

5 

3.71572 

3.34565 

3.65676 

3-40999! 

3.59669 

3-47329;    5 

60 

4.45886 

4.01478 

4.38812 

4.091991 

4.31603 

4.16795!    6 

7 

5.20201 

4.68391 

5.H947 

4.77398     5.03537 

4.86260     7 

8 

5.94515 

5.35304 

5.85082 

5.45598 

5.75471 

5.55726!    8 

9 

6.68830 

6.02217 

6.58218 

6.13798 

6.47405 

6.25192     9 

i 

0.74021 

0.67236 

0.72837 

0.68518 

0.71630 

0.69779     i 

2 

1.48043 

1-34473 

1.45674    1.37036      1.43260 

I.3955S       2 

3 

2.22065 

2.01710 

2.185II 

2.05554;  2.14890 

2.09337 

3 

4 

2.96087 

2.68946 

2.91348 

2.74073  ;  2.86520 

2.79116  i  4 

15 

5 

3.70109 

3-36183 

3.64185    3.42591  !   3.58I5I 

3-48895;;  5 

45 

6 

4.44130 

4.03420 

U.37022    4.11109! 

4.29781 

4.18674 

6 

7 

5.18152 

4.70656 

5.09859    4.79628     5.01411 

4.88453 

7 

8 

5.92174 

5.37893 

5.82696 

5.48146   5-73041 

5-58232 

8 

9 

,6.66196 

6.05130 

6-55533 

6.16664 

6.44671 

6.28011 

9 

i     0.73727 

0.67559 

0.72537 

0.68835  ' 

0.71325 

o.  70090  !|  i 

2 

r.  47455 

I.35H8 

:  i.  45074   1.37670    1.42650 

I.40I8I       2 

3 

2.21183 

2.02677 

2.I76I2      2.06506     !   2.13975 

2.  IO272 

3 

4 

2.94910 

2.70236 

2.90149      2.75341    i     2.85300 

2  .  80363     4 

30 

5 

3.68638 

3-37795 

3.62687      3.44177        3.56625 

3.50454     5 

30 

6 

;  4-  42366 

4-05354 

4.35224 

4.13012  14.27950 

4.20545 

6 

7 

5.16094 

4.72913 

'5.07762 

4.81848   4-99275 

4.90636 

7 

8 

5.89821 

5.40472 

5.80299 

5  .  50683     5  .  70600 

5.60727     8 

9 

6.63549 

6.08031 

16.52836 

6.19519 

6.41925 

6.30818  |  9 

| 

i 

0.73432 

0.67880 

0.72236      0.69I5I        0.7IOI8 

0.70401     i 

2 

;  i.  46864 

i.3576o 

1.44472      1.38302 

1.42037 

1.40802       2 

3 

2  .  20296 

2.03640 

'  2.I6/O9 

2-07453 

2.13055 

2.  II2O4 

3 

4 

2.93729 

2.71520 

2.88945 

2.76605 

2.84074 

2.81605 

4 

45 

.5 

3.67161 

3.39400 

3.6II82      3.45756 

3.55092 

3-52007:    5 

15 

6 

4.40593 

4.072801:4.33418   4.14907 

4.26111 

4.22408;    6 

7 

5.14025 

4.75i6o 

5.05654 

4.84059 

4.97129 

4.92810 

7 

8 

5.87458 

5.43040 

5.77891       5.53210 

5.68148 

5.63211 

O 

9 

6.60890 

6.  10920 

6.5OI27 

6.22361 

6.39166 

6.33613 

9 

g 

5' 

S       Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.     2 

3" 

c 

*r 

;    ^3 

c 

? 

47" 

46°                            45°                  8 

0 

en 

44 


TRAVERSE    TABLE. 


Differences  of  Latitude  and  Departures  —  Continued. 

. 

45° 

Lat. 

Dep. 

I 

o.  70710 

o.  70710 

i 

2 

1.41421 

1.41421 

2 

3 
4 

2.  12132 
2.  82842 

2.  I2I32 
2.  82842 

3 

4 

5 
6 

3-53553 
4.  24264 

3-53553 
4,  24264 

5 
6 

• 

7 
8 

9 

4-  94974 
5.65685 
6.36396 

4-  94974 
5.  65685 
6.  36396 

7 
8 

9 

Dep. 

Lat. 

45° 

Chains,  Yards,  and  Feet, 

WITH   THEIR    RECIPROCAL   EQUIVALENTS. 

Link  —  7.  92  inches.      Chain  =  66  feet  =  792  inches. 


CHAINS    INTO  FEET. 

FEET  INTO   LINKS. 

c/3 

"3 

(3 

Yards. 

Feet. 

Feet. 

Yards. 

Links. 

CJ 

a 

o 

I 

0.22 

0.66 

O.  IO 

•°33 

0.15 

o 

2 

0.44 

1.32 

o.  20 

.066 

o.  30 

o 

3               o.  66 

1.98 

0.25 

.082 

0.38 

o 

4 

0.88 

2.64 

o.  30     i          .  oio 

0-45 

o 

5 

I.  IO 

3.30 

o.  40 

•133 

o.  60 

o 

6 

1.32 

3-96 

o.  50 

.166 

o.  76 

o 

7 

1-54 

4.62 

o.  60 

.  200 

0.91 

o 

8 

1.76 

5.28 

o.  70               .  233 

1.  06 

o 

9 

1.98 

5-94 

0-75 

.250 

1-13 

o 

IO                    2.  2O 

6.60 

o.  80               .  266 

I.  21 

MISCELLANEOUS. 


45 


Chains,    Yards,   and  Feet  —  Continued. 

CHAINS    INTO  FEET. 

FEET   INTO     LINKS. 

'rt          *? 

Yards. 

Feet. 

Feet. 

Yards. 

Links. 

6  a 

0           20 

4-40 

13.  20 

0.9 

-3° 

1.36 

o        30 

6.60 

19.80 

1.0 

•33 

I-5I 

o        40 

8.80 

26.  40 

2.0 

.66 

3-o 

o        50 

II.  OO 

33-  °° 

.  3-° 

I.  OO 

4-5 

o        60 

13.20 

39.  60               4.  o 

i-33 

6.0 

o         70 

15.40 

46.20               5.0 

1.66 

7-5 

o        So 

17.  60 

52.  So    ;           6.  o 

2.00 

9.1 

o        90 

19.80 

59-40 

,7-° 

2-33 

10.6 

I  •         OO 

22.  OO 

66.  oo               8.  o 

2.66 

12.  I 

2           00 

44-00 

132                     9.0 

3.00 

I3.6 

3 

66 

198                    10 

3-  33 

15.  I 

4 

88            •    264                   15 

5.00              22.7 

5 

IIO 

330                           20 

6.66 

30.3 

6 

132 

396                            24 

8.00 

36.3 

7 

J54 

462 

27 

9.00 

40.9 

8 

176 

528                          30 

IO.  OO 

45-4 

9 

198 

594 

33 

II.  OO 

50.  o 

10                                220 

660                  36 

12.  OO 

54-5 

20                                440 

1320 

39 

13.00 

59-  i 

30 

660 

1980 

40 

13-33 

60.6 

35 

770 

2310 

42 

14.00 

63-3 

4° 

880 

2640 

45 

15.00 

68.2 

45 

990 

2970 

48 

16.  oo 

72.7 

5° 

IIOO 

33°° 

5° 

16.66 

•75-7 

55 

1210 

363° 

51 

17.00 

77-3 

60 

1320 

3960 

54 

18.00 

81.8 

65 

143° 

4290 

57 

19.  oo 

86.3 

70 

1540 

4620 

60 

20.  oo 

90.  9 

75 

1650 

4950 

63 

21.  OO 

95-4 

80                  i  760 

5280 

66 

22.  OO 

IOO 

46"  RAILROAD    CURVES. 


XIX. — To  trace  Railroad  Curves  by  means  of  Deflections. 

GENERAL    PROPOSITIONS. 

1.  The  angle  formed  by  a  tangent  and  a  chord  is  equal  to  half 
the  angle  at  the  center  of  the  circle  subtended  by  the  chord. 

2.  The  angle  of  deflection  formed  by  any  two   equal  chords 
meeting  at  the  circumference  is  equal  to  the  angle  at  the  center, 
subtended  by  either  cord. 

3.  A  line  bisecting  the  angle  of  deflection  formed  by  any  two 
equal  chords  is  a  tangent  to  the  arc  at  the  point  where  the  two 
chords  meet. 

4.  If  an  arc  of  a  circle  be  subdivided  into  any  number  of  equal 
parts,  and  lines  be  drawn  from  the  several  points  of  subdivision 
so   as  to  meet  at  any  point  in  the  circumference,  these  several 
lines  will  form   equal  angles   at  the  point  of  meeting,  and  the 
angles  thus  formed  will  be  respectively  measured  by  one-half  the 
subdivided  arc. 

METHOD    BY    DEFLECTION-ANGLES.     * 

The  degree  of  a  curve  is  determined  by  the  angle  subtended  at 
its  center  by  a  chord  of  100  feet. 

The  deflection-angle  of  a  curve  is  the  acute  angle  formed  at  any 
point  between  a  tangent  and  a  chord.  It  is,  therefore,  half  the 
degree  of  the  curve. 

In   order  to  unite  two  straight  lines  by  a  curve,  the  angle  of 
intersection  is  measured,  and  then  a  radius  for  the  curve  may  be 
assumed  and  the  .  tangent  calculated,  or  the  tangent  may   be 
assumed  of  a  certain  length  and  the  radius  calculated. 
Let  I  =  angle  of  intersection  of  the  two  lines; 
R  =  radius  of  circle; 

T  =  length  of  tangent,  or  distance  from  point  of  inter 
section   to  point  where   the  curvature  is  to 
commence;   and 
D  =  angle  of  deflection. 
Then 

T  =  R  tan  J-  I 
R  =  T  cot  \  I 

sin  D=i0  =  5°_tanJ_I 
R  I 


RAILROAD    CURVES. 


47 


XIX. — To  trace  Railroad' Currcs,  erY. — Continued. 

To  lay  out  a  curve,  set  the  instrument  at  the  point  at  which 
the  curvature  is  to  commence,  lay  off  the  given  deflection-angle, 
and  the  first  point  in  the  curve  will  be  at  the  end  of  100  feet 
measured  on  this  new  direction. 

Then  lay  off  another  deflection-angle  equal  to  the  first;  attach 
the  loo-foot  chain  to  the  point  last  found,  and  swing  it,  stretched, 
until  its  extremity  intersects  tile  new  direction,  which  will  be  the 
second  point;  and  so  on.  Should  it  be  found  necessary  to  remove 
the  instrument  from  its'  first  position,  either  on  account  of  the 
length  of  the  curve  or  of  some  obstruction  to  the  sight,  the  first 
deflection  at  the  new  position  of  the  instrument  will  be  equal  to 
the  total  deflection  from  the  preceding  position. 

METHOD  BY  TANGENT  AND  CHORD  DEFLECTION. 

Tangent-deflection  is  the  distance  between  the  extremities  of  a 
tangent  and  a  chord,  each  100  feet  long. 

Chord-deflection  is  the  distance  from  the  extremity  of  the  first 
chord,  produced  an  additional  100  feet,  to  the  extremity  of  the 
next,  and  is,  therefore,  double  the  tangent-deflection. 

To  lay  out  a  curve,  stretch  the  loo-foot  chain  from  the  point 
of  beginning  in  the  direction  of  the  tangent,  and  mark  its  extrem 
ity  ;  swing  the  chain  toward  the  direction  of  the  curve,  keeping 
trie  initial  point  fixed,  until  it  has  diverged  a  distance  equal  to  the 
tangent-deflection ,  which  will  be  \\\e  first  point  of  the  curve. 

Produce  the  first  chord  an  additional  100  feet,  and  swing  the 
chain  (round  the  extremity  of  the  first  chord  as  a  pivot)  until  it 
has  diverged  a  distance  equal  to  the  chord-deflection,  which  will 
be  the  second  point  of  the  curve. 

Continue  to  lay  off -the  chord-deflection  from  the  preceding 
chord  produced  until  the  curve  is  finished. 

LENGTH    OF   CIRCULAR   ARCS    IX   PARTS    OF    RADIUS. 


0 

/ 

;/ 

I 

.01745    32925 

i 

.00029    08882 

.00000    48481 

2 

.03490    65850 

2 

.00058    17764 

.  ooooo    96962 

3 

.05235    98775 

3 

.00087    26646 

.00001    45444 

4 

.  0698  i    31  700 

4 

.00116    35528 

.•ooooi    93925 

5 

.08726    64625 

5 

.00145    444!0 

.00002    42406 

6 

.10471    97551 

6 

.00174    53292 

6 

.00002      90888 

7 

.12217    30476 

7 

.00203    62174 

7 

.00003     .39369 

8 
9 

.13962    63401 
.15707    96326 

8 
9 

.00232    71056 
.00261    79938 

8 
9 

.OOOO3      87850 
.00004      36332 

48                                                    RAILROAD    CURVES. 

XIX.  —  To  trace  Railroad  Curres,.&c.  —  Continued. 

,j 

ORDINATES. 

c 

o 

c 

"o 

Q 

<u 

c 

"o 

'  Degree. 

Radii. 

To  circular  arcs  on  a  chord  of  100  feet. 

o 

12$ 

25 

371   • 

5° 

o 
t/3 

rt 

r-1 

O 

•   '     Fat. 

o     5   :  68754.94 

.008 

.  014 

.017 

.018 

.073 

•145 

io      34377.48 

.Ol6 

.027 

.034 

.  036 

.145 

.291 

15      22918.33 

.024 

.041 

.051 

•055 

.218 

.436 

20       17188.76 

.032 

•055 

.068 

.073 

.  291 

.582 

25     I3751.02 

.  040 

.068 

.085 

.  091 

.364 

.727 

30  :  11459-19 

.048 

.082 

.  102 

.  109 

-436 

.873 

35  !    9822.  18 

•056 

•°95 

.  119 

.127 

.509 

i.oiS 

40       8594.  41 

.  064 

.  109 

.136 

•  145 

.582 

i.  164 

45 

7639.  49 

.072 

.123 

.  T53 

.164 

-654 

1.309 

50      6875.  55 

.080 

.136 

.170 

-.182 

.727 

1-454 

55 

6250.51 

.087 

.150 

.187 

.  200 

.800 

i.  600 

i    o      5729-65 

•095 

.164 

.205 

.218 

•873 

1-745 

5       5288.92 

•  103 

.177 

.222 

.236 

•  945 

1.891 

10  !    4911.  15 

.  Ill 

.191 

.239 

•255 

i.  018 

2.  036 

15 

4583.  75 

.119 

.205 

.256 

•  273 

i.  091 

2.182 

20 

4297.  28 

.127 

.218 

.273 

.291 

i.  164 

2.327 

25 

4044.51 

.135 

.232 

.290 

.309 

1.236 

2.472 

30 

3819.83 

.143 

.245 

.307 

•327 

1.309 

2.618 

35       3618.80 

.  151 

.259 

.324 

•  345 

1.382 

2.763 

40 

3437.87 

.159 

.273 

.341 

.364 

1.454 

2.909 

45 

3274.17 

.167 

.286 

.358 

.382 

1.527 

3-054 

5° 

3125-36 

.175 

.300 

.375 

.  400 

i.  600 

3.  200 

55 

2989.48 

.183 

•  3H         .392 

.418 

1.673 

3-345 

2      0 

2864.  93 

.191 

.327         -409 

.436 

1-745 

3.490 

5 

2750.35 

.199 

.  341          .  426 

•455 

i.SiS 

3-636 

10 

2644.  58 

.207 

•355 

.443 

•  473 

1.891 

3-  78i 

15 

2546.  64 

•215 

.368 

.460 

.491 

1.963 

3-927 

20 

2455-  7o 

.223 

.382 

•  477 

•  509 

2.  036 

4.  072  • 

25 

2371.04 

.231 

•395 

•  494 

.527 

2.  109 

4.218 

30 

2292.01 

.239 

.409 

•  511 

•545 

2.  iSl 

4-363 

35 

2218.09 

•247 

.423 

.528 

064 

2.254 

4.508 

40 

2148.79 

.255 

.436 

•  545 

.582 

2.327 

4-654 

45. 

2083.  68 

•263 

•  45° 

.562 

.600 

2.  4OO 

4-799 

50       2022.41 

.270 

.464 

.580 

.618 

2.472 

4-945 

55       1964.64 

.278 

•  477 

•597         .636 

2.545             5.090 

RAILROAD    CURVES. 


49 


XIX.  —  To  trace  Railroad  Curves,  £rv.  —  Continued. 

ORDINATES. 

ci 

o 

c 

Q 

Degree. 

Radii. 

To  circular  arcs  on  a  chord  of  100  feet.  J       ^ 

"o 
V 

•p! 

"f 

!                                                                     O 

t/3 

f 

124- 

25 

37i 

5°             § 

0 

EH 

0 

o      / 

3     ° 

1910.08 

.286 

.491 

.614 

.655 

2.618 

5-235 

5 

1858.47 

.294 

•505 

.631 

.673 

2.690 

5.381 

10 

1809.57 

.302 

.518 

.648 

.  691 

2-  763 

5.526 

15 

1763.18 

.310 

•532 

•  665 

.709 

2.836 

5.672 

20 

1719.  12 

.318 

•545 

.682 

.727 

2.908 

5-817 

25 

1677.20 

.326 

•559 

.699 

•    -745 

2.981 

5.962 

3° 

1637.28 

.334 

•573 

.716 

.764 

3-  °54 

6.108 

35 

I599-2I 

.342 

.586 

•733 

.782 

3.127 

6.253 

40 

1562.88 

•35° 

.600 

•  75° 

.800 

3.199 

6.398 

45 

1528.  16 

.358 

.  614 

.767 

.818 

3.272 

6.544 

5° 

1494-  95 

.366 

.627 

-784 

.836 

3-345 

6.689 

55 

1463.  16 

•  374 

.641 

.801 

•855  !|     3.4I7 

6.835 

4-     o 

1432.  69 

.382 

.655 

.818 

'    -873 

3.490 

6.  980 

5 

1403.  46 

.390 

.668 

.835 

.891 

3-563 

7.125 

10 

I375-4Q 

.398 

.682 

.852 

.909 

3.635 

7.271 

15 

1348.45 

.406 

.695 

.  869 

.927 

3.708 

7.416 

20 

1322.53 

.414 

.709 

.886 

•945 

3-78I 

7-561 

25 

1297.58 

.422 

•723 

.903 

.  964 

3-853 

7./07 

3° 

1273.57 

.430 

.736 

.921 

.982 

3-926 

7.852 

35 

1250.42 

.438 

.750 

.938 

I.  000 

3-999 

7-997 

40 

1228.  ii 

.446 

.764 

•955 

1.018 

4.071 

8.143 

45 

1206.  57 

•454 

•777 

•972 

1.036 

4.144 

8.288 

5° 

1185.78 

.462 

.791 

.989 

'•055  \ 

4.217 

8-433 

55 

1165.  70 

.469 

.805 

i.  006 

1.073 

4.289 

8-579 

5    ° 

1146.28 

•477 

.818 

1.023 

i.  091 

4.362 

8.724 

5 

1127.50 

-485 

.832 

i.  040 

1.  109  ' 

4.435 

8.869 

10 

1109.33 

.493         -846 

1.057 

1,127 

4.507 

9.014 

15 

1091.  73 

.501         .859 

1.074 

i.  146 

4.580 

9.  1  60 

20 

1074.  68 

•5°9 

.873 

1.091 

1.  164 

4.653 

9-  3°5     . 

25 

1058.  16 

•5*7 

.887 

1.  108 

I.   182     ; 

4.725 

9-45° 

3° 

1042.  14 

•525 

.900 

1.125 

1.200 

4.798 

9.596 

35 

1026.  60 

•533 

.914 

1.  142 

1.218  , 

4.870 

9.741 

40 

1011.51 

.541 

.928 

I-I59 

I-237 

4-943 

9.886 

45 

996.  87 

.549 

.941 

i.  176 

1.255  ;     5.016 

10.031 

5<3                                                   RAILROAD    CURVES. 

XIX.  —  To  trace  Railroad  Curves,  &c.  —  Continued. 

ORDINATES. 

.2 

G 

•g 

_0 

tC 

3    ' 

Degree. 

Radii. 

To  circular  arcs  on  a  chord  of  100  feet,  j 

*"O 

u 

.* 

25 

37* 

SO      I 

<u 
fcJD 
C 
rt 

H 

o 

6 

0           / 

5  50 

982.  64  ! 

-557 

.955 

i.i93 

1.273 

5.088 

10.177 

55 

968.  81 

-565 

.968 

I.  2IO 

1.291  ;•: 

5.  161 

10.322 

6    o 

955-37 

•  573 

.982 

1.228 

1.309   ' 

5-234 

10.467 

5 

942.  29 

.581 

.996 

1.245 

1.327 

5-306 

10.  612 

10 

929-57  ! 

.589 

i  .  009  . 

1.262 

1.346 

5-379 

10.  758 

15 

917.19 

-597 

1.023 

1.279 

1.364 

5-451 

10.  903 

20 

905.  13  ; 

.605 

1.037 

1.296 

1.382 

5.524 

11.048 

25 

893.395     -613 

1.050 

I-3I3 

1.400 

5-597 

11.193 

3° 

881.95  ||    .621 

1.064 

1.330 

1.418 

5.669 

11.339 

35 

870.  79  j 

.629 

1.078 

1.347 

1-437   ! 

5-742 

11.484 

40 

859.92 

.637 

i.  091 

1.364 

1-455 

5-814 

II.  629 

45 

849-  32         .  645 

1.  105 

1.381 

1-473 

5.887 

11.774 

5° 

838.  97  i       •  653 

i.  118 

1.398 

1.491  i 

5.960 

11.919 

55 

828.88         .661 

1.132 

i.4i5 

1.510 

6.  032 

12.065 

7    ° 

819.02         .669 

i.  146 

1.432 

1.528 

6.  105 

12.210 

5 

809.  40 

.677 

I.I59 

1.449 

1.546 

6.177 

12-355 

10 

800.  oo 

.685 

I.I73 

1.466 

1.564 

6.250 

12.  500 

15 

790.81 

•693 

i.  187 

1.483 

1.582 

6.323 

12.645 

20 

781.84  !       .  701 

1.200 

1.501 

i.  600 

6-395 

12.  790 

25 

773-  °7         •  709 

I.2I4 

I.5I7 

i.  619 

6.468 

12.936 

3° 

764.  49 

.717 

1.228 

1-535 

1.637 

6.540 

I3.08I 

35 

756.  10          .  725 

1.242 

1-552 

1.655 

!       6.613 

13.226 

40 

747-  89 

-733 

1.255 

1.569 

1.673 

'  6.  685 

I3.37I 

45 

739-  86 

.740 

I.  269 

1.586 

i.  691 

6.758 

13.516 

•       5° 

732.01 

.748 

1.283 

i.  603 

i.  710 

6.831 

13.661 

55 

724.31 

.756 

1.296 

i.  620 

1.728 

6.903 

13.  806 

8    o 

716.  78 

.764 

I.3IO 

1.637 

1.746 

6.976 

13.951 

5 

7°9-  40         .  772 

1.324 

1.654 

1.764 

7.048 

14.  096 

10 

702.  1  8 

.780 

1-337 

1.671 

1.782 

7.  121 

14.241 

15 

695-  09 

.788 

I.35I 

i.  688 

1.801 

7-  J93 

14.387 

20 

688.  16 

-796 

I.  365 

i.  705 

1.819 

7.266 

14.  532 

25 

681.35 

.804 

1.378 

1.722 

1.837 

7.338 

14.677 

3° 

674.  69 

.812 

1.392 

1.739 

1.855 

7.411 

14.  822 

35 

668.  15 

;         .820 

1-757 

1.873 

7-483 

14.  967 

RAILROAD    CURVES.                                                    51 

XIX.  —  To  trace  Railroad  Curves,  &c.  —  Continued. 

ORDIXATES. 

d 
o 

"o 

,0 

Degree. 

Radii. 

To  circular  arcs  on  a  chord  of  100  feet. 

o 

1 

12* 

25 

371 

5° 

5    . 

o 

H 

o 

8  40 

661.  74 

.828 

1.419 

1-774 

1.892 

7.556 

15.  112 

45 

655.45 

.836 

L433 

1.791 

i.  910 

7.628 

15.257 

50 

649.  27 

.844 

1.447 

i.  808 

1.928 

7.701 

15.402 

55 

643.  22 

.852 

1.460 

1.825 

1.946 

7-773 

15.547 

9    o 

637.27 

.860 

1.474 

1.842 

1.965 

7.846 

15.692 

5 

631.44 

.868 

1.488 

1.859 

1.983 

7.918 

15.837 

10 

625.  71 

.876 

i.  501 

1.876 

2.  OOI 

7.991 

15.982 

15 

620.  09 

.884 

L5I5 

1.893 

2.  OI9 

8.063 

16.  127 

20 

614.  56 

.892 

1.529 

i.  910 

2.037 

8.136 

16.272 

25 

609.  14 

.  900 

1.542 

1.927 

2.  056 

8.208 

16.417 

30 

603.  80 

.908 

1.556 

1.944 

2.074 

8.281 

16.  562 

35 

598.57 

.  916 

I-57° 

1.961 

2.  092 

8-353 

16.  707 

40 

593-42 

.924 

1-583 

1.979 

2.  no 

8.426 

16.  852 

45 

588.  36 

.932 

1.597 

1.996 

2.  128 

8.498 

1  6.  996 

5° 

583-38 

.940 

i.  6n 

2.013 

2.147 

8.571 

17.  141 

55 

578.49 

.948 

i.  624 

2.  030 

2.  165 

8.643 

1  7.  286 

10      0 

573.69 

.956 

1.638 

-  2.  047 

2.183 

8.716 

I7.43I 

10 

564.31 

•  972 

1.665 

2.  081 

2.  219 

8.860 

17.721 

20 

555.23 

.988 

1.693 

2.115 

2.  256 

9.005 

18.  on 

3° 

546.44 

1.004 

1.720 

2.149 

2.292 

9.150 

1  8.  300 

40 

537.92 

i.  020 

1.748 

2.  184 

2.329 

9-295 

18.  590 

5° 

529.  67 

1,036 

1-775 

2.  2l8 

2.365 

9.440 

18.880 

II      0 

521.67 

1.052 

1.802 

2.252 

2.4O2 

9.585 

19.  169 

10 

5I3.9I 

i.  068 

1.830 

2.286 

2.438 

9.729 

19.459 

20 

506.38 

1.084 

1-857 

2.320 

2-475 

9.874 

19.  748 

30 

499.  06 

I.  IOO 

1.884 

2.354 

2.5II 

10.  019 

20.  038 

40 

491.96 

i.  116 

1.912 

2.389 

2-547 

10.  164 

20.  327 

50 

485.  05 

1.132 

1.938 

2.423 

2.584 

10.  308 

20.6l6 

12      O 

478.  34 

i.  148 

1.967 

2-457 

2.  620 

io.453 

2O.  906 

IO 

471.81 

i.  164 

1.994 

2.491 

2.657 

10.  597 

21.  195 

2O 

465.46 

1.  180 

2.021 

2.525 

2.693 

10.  742 

21.484 

30 

459-  28 

1.  196 

2.049 

2.560 

2.730 

10.  887 

21.  773 

40 

453-  26 

I.  212 

2.076 

2-594 

2.766 

11.031 

22.  063 

50 

447.40 

1.228 

2.  104 

2.6«8 

2.803 

n.  176 

22.352 

52 


RAILROAD    CURVES. 


XIX. To  trace  Railroad  Curves,  &c. — Continued. 


ORDINATES. 

. 

0 

.2 

"o 

o 

Degree. 

Radii. 

To  circular  arcs  on  a  chord  of  I  oo  feet.         £ 

i 

.* 

25 

37* 

bX) 

50     :      § 

o 

6  ' 

0           1 

441.68         1.244 

2.131 

2.662 

2.839         11.320 

22.  641 

IO 

436.  12    1        1.260 

2.159 

2.697 

2.876     i    11.465 

22.  930 

20 

430.69           1.277 

2.186 

2.731 

2.912    :    11.609 

23.219 

3° 

425.40     |       1.293 

2.213 

2.765 

2.949  ;  11.754 

23.  507 

40 

420.23           1.309 

2.  241 

2.799 

2.985  j  11.898 

23.  796 

5° 

415.19           1.325 

2.268 

2-833 

3.022     !    12.043 

24.  085 

14    o 

4IO.  28 

1.341 

2.  296 

2.868 

3.  058    ij    12.  187 

24-  374 

10 

405-  47 

1.357 

2.323 

2.902 

3.095    jj    12.331 

24.  663 

20 

400.  78 

1.373 

2.351 

2.936 

3.I3I         12.476 

24.951 

3° 

396.  20 

1.389 

2.378 

2.970 

3.  1  68   1  12.620 

25.  240 

40 

391.  72 

1.405 

2.406 

3-005 

3.204    :   12.764 

25-528 

50 

387-  34 

1.421 

2.433 

3.039 

3.241       12.908 

25.817 

15     o 

383.  06 

1.437 

2.461 

3-073 

3-277 

13.053 

26.  105 

10 

378.88 

1-453 

2.488 

3.107 

3-3H 

13-  197 

26.  394 

20 

374-  79 

i.  469 

2.515 

3-  142 

3-350 

13-341 

26.  682 

3° 

370.  78 

1.486 

2-543 

3.176 

3-387 

13.485 

26.  970 

40 

366.  86 

1.502 

2.570 

3.210 

3.423 

13.629 

27.258 

5° 

363.  02 

1.518 

2.598 

3-245 

3.460 

13.  773 

27.547 

16    o 

359-  26 

1-534 

2.625 

3-279 

3.496 

13-917 

27.835 

IO 

355-59 

1-55° 

2.653 

3.313 

3-533 

14.  061 

28.  123 

20 

35I-98 

1.566 

2.680 

3-347 

3.569 

14.205 

28.411 

3° 

348.  45 

1.582 

2.708 

3-382 

3.606 

14-349 

28.  699 

40 

344-  99 

1.598 

2.736 

3.416 

3-  643 

14-  493 

28.  986 

341.60 

1.615 

2.763 

3.450 

3-679 

14-637 

29-274 

17    o 

338.27 

i  .  63  1 

2.791 

3.485 

3.716 

!  H.  781 

29.  562 

10 

335-  OI 

1.647 

2.818 

3.519 

3-752 

j  14-925 

29.  850 

20 

331-82 

1.663 

2.846 

3.553 

3-789 

15.  069 

30.  137 

3° 

328.  68 

1.679 

2.873 

3.588 

3-825 

15.212 

3°.  425 

40 

325.  60 

1.695 

2.901 

3.622 

3.862 

15.356 

30.  712 

5° 

322.  59 

1:711 

2.  928 

3.656 

3.898 

15.500 

31.  ooo 

18    o 

319.62 

i.  728 

2.956 

3.691 

3-935 

15.643 

31.287 

IO 

316.71 

1.744 

2.983 

3.725 

3-972 

15.  787 

31.574 

20 

313-86 

i.  760 

^.  01  1 

3-759 

4.008  i!  I5-931 

31.861 

RAILROAD    CURVES.                                                    53 

XIX.  —  To  trace  Railroad  Curves,  &c.  —  Continued. 

ORDIXATES. 

*J3 

a 

i                                                                                                   ^ 

o 

1 

! 

Degree. 

Radii.      j.To  circular  arcs  on  a  chord  of  100  feet.  ;        ^ 

\\                                                                            ~Z 

1 

i2|           25            37}- 

bfl 

50 

o 

.  —  1 

|| 

H 

CJ 

o 

18  30 

311.06        1.776       3.039       3.794 

4-045      16.074 

32.  149 

40 

308.30      ;              1.792                  3.066                  3.828 

4.081       16.218 

32.436 

50 

3O5.6O      \              1.809                 3.094                 3.862 

4.118      16.3^1 

32.  723 

19    o 

302.Q4                    1.825                  3.I2I                   3.897 

4.155      16.505 

33.010 

IO 

300.33                     I.84I                   3.    149                  3.931 

4.  191       16.648 

33.  296 

20 

297-77        1.857       3-I77      3-965 

4.228      16.792 

33.583 

30 

295.25  i     1.873       3.204      4.000 

4.265      16.935 

33.870 

40 

292.77  .     1.890       3.232       4.034 

4.301       17.078 

34.157 

5° 

•    290.33  <i     1.906       3.259       4.069 

4.338      17.222 

34-  443 

LONG   CHORDS. 

Degree  : 
of  curve. 

2  stations. 

3  stations. 

4  stations. 

5  stations. 

6  stations. 

0    10 

2OO.  000 

299.  999 

399-  998 

499-  996 

599-  993 

20 

199.  999 

•997 

•992 

.983                .970 

3° 

.998 

.992 

.981 

.962 

•933 

40 

•997 

.986 

.966 

•932 

.882 

50 

•995 

•979 

•947 

•894 

.815 

I       0 

199.  992 

209.  970 

309.  924 

499.  848 

599-  733 

IO 

.990 

•  959 

.896 

.    -793 

.637 

20 

.986 

.946 

.865 

.729 

.526 

3° 

•  983 

•  932 

.829 

.657 

.401 

40 

•979 

•915 

.789 

'  -577 

.  260 

50 

.974 

.898 

•744 

.488 

•   -105 

2      0 

199.  970 

209.  878 

399-  695 

499-  39i 

598.  934 

IO 

.964 

•  857 

.643 

.285 

•750 

20 

•959 

•  834 

.586 

.171 

•55° 

30 

•  952 

.810 

•  524 

.049 

.336 

4° 

.946 

.783 

•459 

498.918 

.  106 

50     : 

•939 

•  756 

.  389                .  778 

597-  862 

54                                                      RAILROAD    CURVES. 

XIX.  —  To  trace  Railroad  Curves,  &c.  —  Continued. 

LONG  CHORDS  —  Continued. 

Degree 
of  curve. 

2  stations. 

3  stations. 

4  stations. 

5  stations. 

6  stations. 

0         / 

3    o 

199.931 

299.  726 

399-  3*5 

498.  630 

597.604 

IO 

.924 

•695 

•  237 

•  474 

•331 

20 

.915 

.662 

•  154 

.309 

•043 

3° 

.907 

.627 

.068 

•136 

596.  740 

40 

.898 

•591 

398.977 

497-  955 

•  423 

50 

.888 

•553 

.882 

•765 

.  091 

4    o 

199.878 

299.513 

398.  782 

497.  566 

595-  744 

IO 

.868 

.471 

.679 

.360 

•  383 

20 

.857 

.428 

•571 

•  145 

.007 

30 

.846 

.383 

•  459 

496.921 

594.617 

40 

.834 

•337 

•343 

.689 

.  212 

50 

.822 

.289 

.223 

•449 

593-  792 

5    o 

199.  Sio 

299.  239 

398.  099 

496.  200 

593.358 

10 

•  797 

.187 

397.  970 

495-  944 

592.  909 

20 

,783 

•  134 

•  837 

.  678 

.446 

30 

.770 

.079 

.700 

.405 

591-968 

40 

.756 

.023 

•  559 

.123 

•  476 

50 

.741 

298.  964 

.413 

494.  832 

590.  970 

6    o 

199.  726 

298.  904 

397-  264 

494-  534 

590.  449 

10 

.710 

•  843 

.  no 

.227 

589.  9J3 

20 

.695 

•  779 

396.  952 

493.912 

.364 

3° 

.678 

.714 

.790 

.588 

588.  800 

40 

.662 

.648 

.623 

•  257 

.  221 

5o 

.644 

•  579 

•453 

492.917 

587.628 

7    o 

199.  627 

298.  509 

396.  278 

492.  568 

587.021 

10 

.  609 

.438 

.099 

.212 

586.  400 

20 

•  591 

•  364 

395-  9l6 

491.847 

585.  765 

30 

•  572 

.289 

.729  !         .474 

•  "5 

40 

•553 

.212 

•  538 

•093 

584-451 

50 

•533 

.•34 

•  342 

490.  704 

583-  773 

8    o 

199.513          298.054 

395-  H2 

490.  306 

583.  08  i 

1 

«                                     . 

GAUGING    OF    RIVERS.  55 

XX. — To  ascertain  the  Discharge  of  Water  in  any  Stream. 

i.  For  practically  gauging  large  rivers  a  locality  is  selected  in 
a  straight  portion  of  the  stream  where  the  water  flows  smoothly 
and  without  obstruction.  A  base-line  about  200  feet  long  is  laid 
out  parallel  to  the  current,  and  the  exact  cross-section  in  front 
of  this  base  is  determined  by  careful  sounding. 

To  obtain  the  discharge,  two  theodolites  are  established,  and 
the  angular  distance  from,  and  the  times  of  transit  past,  each  end 
of  the  base,  of  numerous  floats,  well  distributed  between  the 
banks,  are  noted. 

The  floats  should  be  made  double,  the  surface-float  being  a 
minute  tin  ellipsoid,  a  piece  of  cork,  or  some  other  small  light 
body,  bearing  a  small  flag.  The  lower  float  may  be  a  large  box 
or  keg  without  top  or  bottom,  kept  upright  by  lead  ballasting;  or 
better,  because  lighter,  two  sheets  of  tin  bent  at  right  angles, 
and  soldered  together  at  the  bend,  so  as  to  make  all  the  angles 
between  the  four  faces  right  angles;  the  essential  conditions  being 
that  the  lower  float  shall  so  greatly  preponderate  in  area  over 
the  upper,  and  shall  be  connected  by  so  fine  a  wire  or  cord,  that 
its  rate  of  movement  will  govern  the  whole  combination. 

The  center  of  the  lower  float  should  be  placed  at  the  mid- 
depth  of  the  stream,  in  each  vertical  plane  of  transit,  because  the 
rate  of  movement  will  then  be  unaffected  by  wind. 

As  it  is  sometimes  troublesome  to  adjust  to  mid-depth  in  the 
different  planes  of  transit,  when  there  is  a  tolerably  uniform  and 
symmetrical  cross-section,  the  average  mid-depth  of  the  river  may 
be  adopted  for  all  the  floats  without  sensible  error. 

If  floats  passing  near  the  surface  are  used,  errors  in  the  com 
puted  discharge  may  be  caused  by  an  ordinary  breeze,  and  as 
these  errors  are  positive  or  negative  according  to  the  direction  of 
the  wind,  discrepancies  may  result  in  the  measurements  of  differ 
ent  days  when  there  is  no  real  variation  in  the  discharge.  All 
this  uncertainty  is  avoided  by  using  mid-depth  floats. 

The  exact  level  of  the  water-surface  on  a  permanent  gauge- 
rod  should  be  carefully  noted  when  the  observations  begin  and 
terminate. 


5  6  GAUGING    OF    RIVERS. 


XX.  —  To  ascertain  the  Discharge  of  Water,  &c.  —  Continued. 

Upon  a  sheet  of  section-paper  the  base-line  and  the  two  per 
pendiculars  across  which  the  times  of  transit  were  noted  are  then 
laid  down,  and,  from  the  recorded  angles  and  a  table  of  natural 
tangents,  the  distances  from  the  base-line  to  the  points  at  which 
each  float  passed  both  lines  are  plotted.  These  points,  being 
connected,  indicate  the  paths  of  the  floats.  Upon  each  path  the 
difference  between  the  two  recorded  times  of  transit  is  written  in 
seconds.  These  seconds  of  transit  are  next  examined,  and  the 
total  width  of  the  river  is  marked  off  into  as  many  "divisions"  as 
it  seems  proper  to  assume  are  traversed  by  water  moving  with 
sensibly  unvarying  veolcity,  say,  for  instance,  that  each  division 
is  about  Jy  of  the  width  of  the  stream.  A  mean  of  the  seconds 
of  transit  of  all  the  floats  in  each  "  division  "  is  next  taken,  and, 
when  reduced  to  velocity  in  feet  per  second,  is  adopted  as  the 
mid-depth  velocity  in  that  "division." 

A  mean  of  all  these  mean  mid-depth  velocities  —  interpolations 
being  made  if  any  are  missing  —  closely  approximates  the  mean 
velocity  of  the  river,  provided  the  "  divisions  "  are  equal  in  width. 

This  method  involves  two  errors,  which  nearly  balance  each 
other,  viz:  the  inequality  in  area  of  the  divisions,  and  the  differ 
ence  between  the  mid-depth  velocity  and  the  mean  velocity  in 
any  vertical  plane,  giving  a  resulting  mean  velocity  of  about  0.95 
times  its  true  value. 

The  mean  velocity  in  each  plane  is  obtained  from  the  mid- 
depth  velocities,  V  J  D,  by  the  formula  — 

7'  =  1.075  Vi  D  +  0.004  £  —  0.093  (V  -J  I),  b)* 
and  — 

b-        T'69 
B{D+i.S)* 

where  — 

D  =  the  depth  of  the  stream  at  any  point  of  the  surface. 

If  a  is  the  area  of  cross-section,  and  a',  a",  etc.,  the  partial 
division  areas,  the  discharge  may  be  found  by  — 


where 


denotes  the  sum  of  similar  quantities. 


GAUGING    OF    RIVERS.  57 


XX.  —  To  ascertain  the  Discharge  of  Water  ,  &c.  —  Continued. 

2.  Determination  of  the  mean  velocity  in  terms  of  the  dimen 
sions  of  the  cross-section  and  the  slope. 

Humphreys-Abbot  Formula. 
(Not  applicable  to  water  flowing  in  smooth  artificial  channels.) 


=      V  0.0081  b+  (225  /-1^)i_  0.09 


!+/ 

where  the  symbols  have  the  following  signification,  all  expressed 
in  English  feet : 

.          v  =  mean  velocity  =-    ; 

a  =  area  of  cross-section ; 
W  =  width ; 

r  =  mean  radius,  or  -  • 

7/  =  value  of  first  term  in  expression  for  T\ 
p  =  wetted  perimeter ; 
Q  =  discharge  in  cubic  feet  per  second; 
a 

=7Tw; 

5-  =  sine  of  slope  of  water  surface  .corrected  for  bends; 

b  =  function  of  the  depth,  for  small  streams  = '—^ 

(r+  1.5)* 

For  rivers  whose  mean  radius  exceeds  12  or  15  feet,  b  may  be 
assumed  to  be  0.1856,  which  will  make  the  numerical  value  of 
the  term  involving  b  so  small  that  it  may  be  generally  neglected, 
reducing  the  above  equation  to — 

»:=([  225^]* -0.0388)' 

The  following  formulae  give  the  value  of  each  variable  in  terms 
of  the  others  and  known  quantities : 

z  =  0.93  v  -f-  0.167  (P  l')*  > 

and  when  -p  is  not  known  by  measurement  it  may,  for  ordinary 
natural  channels,  be  assumed  to  be  1.015  ^. 


+  W)*»\ 
195  a      ) 


a  =  (P  +  W)  z* 

'95  W* 
p  +  W  =  T95      (^ 


58  GAUGING    OF    RIVERS. 


XX. —  To  ascertain  the  Discharge  of  Water,  err. — Continued. 

APPLICATION. — The  variables  which  enter  these  formulas  require 
a  knowledge  of  the  mean  cross-section  of  the  stream,  and  a  map 
of  the  course  of  the  channel  between  two  selected  points  of  the 
water-surface,  whose  difference  of  level  should  be  exactly  known. 

Whenever  practicable,  the  two  points  should  be  located  on  a 
straight  and  regular  portion  of  the  river  to  eliminate  the  effects  of 
bends.  As  this  is  not  always  possible,  the  general  case  is  con 
sidered 'in  the  above  fomiulce,  and  bends  are  assumed  to  exist 
between  the  points  selected. 

The  field-operations  consist  in  a  survey  of  the  channel,  with 
numerous  soundings  between  permanent  bench-marks  placed  near 
the  water,  and  in  running  a  line  of  levels  between  those  marks, 
so  as  to  give  their  relative  level  with  the  most  extreme  accuracy. 

These  points  should  be  located  with  care,  as  far  apart  as  prac 
ticable,  distant  from  any  eddy,  and  placed  where  the  current  on 
the  banks  flows  with  equal  velocity.  This  latter  condition  is  neces 
sary,  because,  as  water  in  motion  exerts  less  pressure  than  when 
at  rest,  if  it  moved  rapidly  past  one  bench-mark  and  was  nearly 
stationary  at  the  other,  a  difference  of  level,  which  has  nothing  to  do 
with  the  motive  power  of  the  stream,  would  vitiate  the  observation. 

In  determining  the  mean  dimensions  of  cross-sections,  care 
must  be  taken  to  extend  the  soundings  throughout  the  entire 
distance  between  the  bench-marks,  and  it  must  be  borne  in  mind 
that  measured  fall  in  water-surface  between  two  stations  corre 
sponds  to  the  mean  channel  between  them. 

When  the  soundings  are  made,  the  water-level  should  be 
referred  to  the  bench-marks  in  order  to  determine  the  area 
corresponding  to  any  subsequent  stand  of  the  river. 

These  soundings  completed,  frequent  gauging  of  the  river  can 
be  made  by  referring  at  any  time,  by  accurate  levels,  the  water- 
surface  at  the  two  points  to  their  respective  bench-marks,  thus 
determining,  the  fall  and  corresponding  cross-section  of  the  river 
from  which  the  discharge  is  computed. 

The  observations  must  be  simultaneous  in  order  to  avoid  the 
effect  of  any  oscillation  in  the  river,  and  calm  days  should  be 
selected  because  waves  render  it  difficult  to  determine  the  exact 
level  of  the  water-surface;  and,  also,  changes  of  level  result  from 
the  general  piling  up  or  lowering  of  the  water  under  the  influence 
of  winds. 


GAUGING    OF    RIVERS. 


59 


XX.  —  To  ascertain  the  Discharge  of  Water  ,  &c.  —  Continued. 

Correction  for  Bends.  —  A  line  following  the  mid-channel  is 
drawn  on  the  map,  composed  of  straight  lines  with  angular 
changes,  wherever  necessary,  of  30°.  A  mean  velocity  is  assumed, 
to  be  corrected  subsequently  if  required,  and  the  value  of  h  is 
computed  in  the  following  formulas,  in  which  N  represents  the 
number  of  deflections  : 

N  sin2  30° 


134 

The  deduced  value  of  h  is  next  subtracted  from  the  total  fall 
in  the  water-surface  between  the  two  stations;  the  remainder 
divided  by  the  distance  in  feet  between  these  stations,  measured 
on  the  middle  line  of  the  river,  is  the  true  value  of  s  in  the 
formula  for  mean  velocity. 

If  any  material  error  has  been  made  in  assuming  r,  the  com 
putation  should  be  repeated  until  the  requisite  approximation  has 
been  made. 

By  expressing  the  formula  in  the  form  — 


M 


—  M'  V 


The  following  table  will  facilitate  its  application: 


r 

M 

I/  At 

P 

M' 

Log  M' 

i     o.  0087 

o.  0930 

5 

o.  400 

9.  602060 

2 

73 

855 

6 

•343 

9-  535294 

-> 

65 

803 

7 

.300 

9.477121 

4 

58 

764 

8 

.267 

9.426511 

c 

54       733 

9 

.  240 

9.  380211 

6 

^o  '       707 

10 

.218 

9-  338456 

7  i        47 

685 

12 

.185 

9.267172 

8 

44 

666 

14 

.  160 

9.  204120 

9 

42 

649' 

16 

.141 

9.  149219 

10 

4° 

634 

18 

.126 

9.  100371 

12 

37 

610 

20 

.114 

9-  056905 

14 

35 

590 

22 

.  104 

9.017033 

.   16 

33 

573 

24 

.  096 

8.982271 

18 

558 

26 

.089 

8.  949390 

20 

29 

544 

28 

.083 

8.919078 

3° 

24 

494 

3° 

.078 

8.  892095 

50 

Q.  0019 

0.0437 

50 

0.047 

8.  672098 

For  streams  larger  than  50  or  100  feet  in  cross-section  the  term 
involving  M'  may  be  dropped,  and  for  larger  rivers,  exceeding  1 2 
or  20  feet  in  mean  radius,  M,  but  not  \l  M,  may  be  neglected. 


6o 


GAUGING    OF    RIVERS. 


0 

3 

0 

0 

G 

Q 

J~ 

QJ 

^ 

Q 

I 

•p3,nduioD 

| 

i 

& 

*o 

u-> 

•*• 

CO 

G 

cS 

0 

o 

•p3Aiosqo 

o 

; 

S 

1            "> 

ro 

ro 

co 

VO 

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$ 

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. 

6 

6 

o 

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«  6 

rt       CL, 

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$ 

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£ 

u 

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</: 

1 

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1 

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0 

? 

^ 

o 

S 

5 

ci 
o 

^     ro     ^*- 

85 

ON 

CO 

CO 

OO 

o 

0 

^ 

'. 

Q 

C 

0 

MM 

i  —  > 

| 

H 

• 

a 

S 

jj 

u 

-; 

"3 

d 

o 

: 
u 

0 

J 

"o 

3 

S 

O 
?> 

Q 

U 

525 

; 

o 

o 

s 

o 

3 

iquemin 

ke&Oh 

feeder. 

8 

1 

E 

3 
O 

- 

1 

3 

O 

S 

K 

MEASUREMENT    OF    FLOWING   WATER.  6 1 


XX. — To  ascertain  the  Discharge  of  Water,  &c. — Continued. 

3.  Formulae  for  the  mean  velocity,  from  other  authorities  : 

f  Downing's  and  others'  co-efficient v  =  100.0  (rs)* 

Chezy . . .  <(   Eytelwein's  co-efficient v  =    93.4  (rs) 

i.  Young's  co-efficient v=    84.3  (rs)' 

!For  canals v  =  (0.0556  +  10593  rs)*  -  0.235 7 
For  canals  and  pipes  ....... v  =  (0.0237  +  9966  rs)*  —  o.  1542 
Eytelwein's  co-efficient .'.... v  =  (0.0119  +  8963  rs)*  —  0.1089 
Weisbach's  co-efficient v  =  (0.00024  +  8675  rs)^  —  0.0154 

/  IOOO  S 

Darcy-Bazin Z/  =  ^0^8534^+^0^5 


XXI.  —  Motion  of  Water  in  Conduit-  Pipes. 

Discharge  through  pipes  of  uniform  dimensions,  and  having 
no  sudden  changes  of  direction  : 

For  ordinary  cases  : 


Q  =  38.436      —-  -  0.070862  D 
In  great  velocities  : 

Q  =  36.7*9 


If  the  velocity  be  required,  divide  the  discharge  by  the   area 
of  the  section  (0.7854  D2). 

To  find  the  diameter  of  a  conduit-pipe  for  a  given  discharge 


under  a  given  head  : 


Where  Q  =  discharge  in  cubic  feet  per  second;  H,  the  head, 
and  D  and  L  the  diameter  and  length  of  the  pipe  in  feet. 

The  resistance  of  curves  is  proportional  to  the  square  of  the 
velocity  of  the  fluid,  to  the  number  of  angles  of  reflextiqn,  and  to 
the  square  of  their  sine, 

Q2 

or,  in  function  of  Q,  =0.006079  —  4  •  j2 

s2  being  the  sum  of  the  squares  of  all  the  sines  of  the  angles  of 
reflexion. 


1 

62                        LOGARITHMS. 

XXII.  —  Logarithms  of  Numbers. 

8 

Proportional  parts. 

^ 

0    1 

2 

3 

4 

5 

6 

y 

8 

9 

fc 

I 

* 

3 

1 

•I 

6 

1 

8 

9 

10 

.0000 

.0043 

.0086 

.0128 

.0170 

.0212 

•  0253 

.0294 

•0334 

•  0374 

4 

8 

12 

'7 

21 

25,29 

33 

37 

II 

.0414 

•°453 

.0492 

•0531 

.0569 

.0607 

.0645 

.0682 

.0719 

•  0755 

4  8 

I  1 

i  g 

19  23  26^30 

34 

12 

.0792 

.0828 

.0864 

.0899 

•°934 

.0969 

.  1004 

.1038 

.  1072 

.1106 

3  7 

IO 

'•1 

17)212428 

31 

T3 

•"39 

•"73 

.I2o6[  .1239  .1271 

•1303 

•  J335 

.1367 

•J399 

.1430 

3  6 

10 

,  ; 

1619232629 

14 

.1461 

.1492 

•  1523 

•1553 

.1584 

.1614 

.1644 

.1673 

.1703 

•1732 

3 

6 

9 

12 

15 

1821 

24 

27 

15 

.1761 

.1790 

.1818 

.1847 

•  1875 

.1903 

•1931 

•1959 

.1987 

.2014 

q 

6 

8 

i  i 

14 

17  20 

22 

25 

16 

.2041 

.2068 

.2095   .2122 

.2148 

•2175 

.2201 

.2227 

•2253 

.2279 

3  5 

8 

i  i 

13  i6'i8  21  24 

17 

.2304 

•  2330 

•2355 

.2380  .2405 

.2430 

•2455 

.2480 

.2504 

.2529 

2  5 

;; 

ii  > 

I2'l5  17  2022 

18 

•2553 

•2577 

.2601 

.2625  .2648 

.2672 

.2695 

.2718  .2742 

.2765 

2  5 

7 

9 

12^4  16  19  21 

19 

.2788 

.2810 

•2833 

.2856 

.2878 

.2900 

.2923 

•  2945 

.2967 

.2989 

2  4 

- 

9  n'i3  16 

IS 

2O 

20 

.3oio|  .3032 

•3054 

•3075 

•  3096 

•  3"8 

•3139 

.3160 

.3181 

•  3201 

2 

4 

< 

8 

II 

13  is 

J7 

19 

21 

.3222 

•3243 

•3263 

•3284  .3304 

•3324 

•3345 

•3365 

•3385 

•  34°4 

2  4 

• 

! 

IOJI2JI4 

(6 

18 

22 

•3424 

•3444 

•3464   0483 

•  3502 

••3522 

•3541 

•3560 

•3579 

•3598 

2  4 

< 

10  12  14 

15 

17 

23 

•36l7 

•  3636 

•3655'  -3674 

.3692 

•37" 

•3729 

•3747 

.3766 

•3784 

2  4 

' 

7 

9"  J3 

15 

17 

24 

.3802  .3820 

•3838 

•3856 

•3874 

.3892 

•3909 

•3927 

•3945 

•  3962 

2. 

4 

1 

9  " 

\  - 

'•i 

16 

25 

•3979  -3997 

.4014 

.4031 

.4048 

.4065 

.4082 

.4099 

.4116 

•4i33 

2 

3 

7 

9  10 

12 

H 

15 

26 

.4150  .4166 

.4183   .4200 

.4216 

.4232 

.4249 

.4265 

.4281  .4298 

2  3 

- 

8  10  ii 

1  .: 

15 

27 

•43J4  -433° 

•4346 

.4362 

•4378 

•4393 

.4409;  .4425 

.4440  .4456 

2  3 

5 

6 

8 

9" 

13 

14 

28 

•4472 

.4487 

.4502 

.4518 

•4533 

•4548 

•4564 

•4579 

•  4594 

.4609 

2  3 

g 

<• 

8 

9" 

12 

«4 

29 

.4624 

•4639 

.4654   .4669 

.4683 

.4698 

.4713 

.4728 

•4742 

•4757 

I 

3 

4 

6 

- 

910 

1  .' 

13 

30 

•4771  .4786 

.4800 

.4814 

.4829 

•4843 

•4857 

.4871 

.4886 

.4900 

I 

3 

6 

7 

I 

II 

,3 

31 

.4914 

.4928 

.4942 

•4955 

.4969 

•4983 

•4997 

.5011 

.5024 

.5038 

I 

3 

i 

7  S'IQ 

II 

12 

S2 

•5051 

•5065 

•5°79 

.5092 

•5I05 

•5"9 

•5132 

•5I45 

•5i59 

•5I72 

'  I 

• 

1 

7  8  9 

II 

12 

33 

0185  .5198 

.5211 

.5224 

•5237 

•5250 

.5263 

.5276 

.5289  .5302 

I 

3 

i 

6|  8 

9  10  12 

34 

•5315 

.5328 

•5340 

•5353 

.5366 

•5378 

•5391 

•54°3 

.5416 

.5428 

I 

3 

• 

6  8 

9 

IO 

II 

35 

•5441 

•5453 

•5465 

•5478 

•549° 

•5502 

•5514 

•5527 

•5539 

•5531 

I 

2 

6  7 

9 

IO 

II 

36 

•5563  -5575 

•5587 

•5599 

.5611 

•5623 

•5635 

•5647 

•5658 

.5670 

I 

2 

g 

6  7 

;- 

IO  II 

37 

.5682 

•5694 

•57°5 

•57I7 

•5729 

•574° 

•5752 

•5763 

•5775 

.5786 

I 

2 

2 

g 

6 

7 

I 

9  10 

38 

.5798 

.5809 

.5821 

•5832 

.5843 

•5855 

.5866 

•5877 

.5888 

•5899 

I 

2 

; 

6 

7 

8 

9  10 

39 

•59" 

.5922 

•5933 

•5944 

•5955 

.5966 

•5977 

.5988 

•5999 

.6010 

I 

9 

! 

- 

7 

8 

.  , 

IO 

40 

.6021 

.6031 

.6042 

.6053 

.6064 

.6075 

.6085 

.6096 

.6107 

.6117 

I 

2 

, 

5 

6 

8 

" 

IO 

41 

.6128 

.6138 

.6149 

.6160 

.6170 

.6180 

.6191 

.6201 

.6212 

.6222 

I 

a 

-i 

i 

( 

7 

' 

9 

42 

.6232 

.6243 

•6253 

.6263 

.6274 

.6284 

.6294 

.6304 

.6314 

•6325 

I 

a 

., 

« 

6 

7 

8 

9 

43 

.6335 

•6345 

•6355 

•6365 

.6375 

•6385 

•6395 

.6405 

.6415 

.6425 

I 

2 

i 

- 

6 

7 

8 

9 

44 

•6435 

.6444 

•6454 

.6464 

.6474 

.6484 

•6493 

.6503 

•6513 

.6522 

I 

a 

3 

! 

-> 

6 

7 

8 

9 

45 

•6532 

•  6542 

•6551 

.6561 

.6571 

.6580 

.6590 

•6599 

.6609 

.6618 

I 

2 

, 

•I 

•-, 

6 

7 

8 

9 

46 

.6628 

.6637 

.6646 

.6656 

.6665 

.6675 

.6684 

.6693 

.6702 

.6712 

1 

2 

, 

s 

6 

7 

7 

8 

47 

.6721,  .6730 

•6739 

.6749 

.6758 

.6767 

.6776 

.6785 

.6794 

.6803 

1 

a 

3 

., 

s 

S 

6 

7 

8 

48 
49 

.6812  .6821 
.6902!  .6011 

.6830 
.6920 

.6839 
.6928 

.6848 

.6857 
6046 

.6866 

.6875 

.6884 

.6893 
.6981 

I 

2 

3 

4 

•1 

5 

6 

7 

8 

50 

.6990 

.6998 

.7007 

.7016 

.7024 

•uy4u 
•7033 

•  ^955 

.7042 

.7050 

.OQ72 

•7°59 

.7067 

z 

9 

i 

s 

... 

7 

8 

51 

.7076 

.7084 

•7°93 

.  7101 

.7110 

.7118 

.7126 

•7*35 

•7*43 

•7152 

I 

2 

3 

; 

r. 

6 

7 

8 

52 

.7160  .7168 

•7177 

•7185 

•7*93 

.7202 

.7210 

.7218 

.7226 

•7235 

I 

2 

.'. 

1 

S 

6 

7 

7 

53 

•7243 

•7251 

•7259 

.7267 

•7275 

.7284 

.7292 

.7300 

.7308 

.7316 

I 

2 

3 

; 

5 

', 

54 

•7324 

•7332 

•7340 

•7348 

•7356 

•7364 

•7372 

•738o 

•7388 

•7396 

I 

2 

a 

1 

S 

6 

67 

_  _ 

LOGARITHMS. 

63 

XXII.  —  Logarithms  of  Numbers  —  Continued. 

0 

Proportional  parts. 

^ 

Q 

-| 

Q 

3 

4 

5 

6 

7 

8 

9 

cd 

1  2 

{ 

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SINES    AND    TANGENTS. 


XXIII. — Logarithms  of  Sines  and  Tangents. 


o° 

i" 

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Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

89° 

88° 

SINES    AND    TANGENTS. 


XXIII. — Logarithms  of  Sines  and  Tangents — Continued. 


2 

3° 

4° 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

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I 

8.5428 
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1.4569 
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30 

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29 

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0 

Cos. 

Sin. 

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Tan. 

Cos. 

Sin. 

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Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

87° 

86° 

85° 

66                      SINES  AND  TANGENTS. 

XXIII.  —  Logarithms  of  Sines  and  Tangents  —  Continued. 

Arc 

Sin. 

Df. 

Cos. 

Df 

Tan.  Df.  j  Cot. 

Arc 

Arc 

Sin. 

Df. 

Cos. 

Df. 

Tan. 

Df. 

Cot. 

Arc 

0  / 

0  / 

0  / 

0  / 

5  o 

8.9403 

1429.9983 

8.9420 

143  1.0580 

85  o 

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9.4130 

47 

9.9849 

3 

9.4281 

50 

0.5719 

75  o 

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137 

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138  .0437 

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46 

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3 

•  433i 

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20 

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20 

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4 

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3° 

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29 

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130  .0164 

3° 

30 

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45 

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3 

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49 

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30 

408.9945 

2  = 

•  9979 

8.9966 

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20 

40 

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50  9.0070 

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30 
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50 

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10  0 

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20 

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3° 

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30 

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40 

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5° 
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3 
1 
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53  1  -6032 
531  -5979 
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450 
40 

24  o 

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20 

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28 
28 
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6 
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40  .4035 

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27 

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9-6687  33 

o-33I3 

650 

Arc 

Cos. 

Df. 

Sin.  Df. 

Cot. 

Df.  Tan. 

Arc 

Arc  Cos. 

Df. 

Sin. 

D£ 

Cot. 

Df. 

Tan. 

Arc 

SINES  AND  TANGENTS. 

67 

XXIII.  —  Logarithms  of  Sines  and  Tangents  —  Continued. 

Arc 

Sin. 

Df. 

Cos. 

Df. 

Tan. 

Df. 

Cot.  Arc 

Arc 

Sin. 

Df. 

Cos. 

I)i 

Tan. 

Df.  Cot. 

Arc 

0  / 

0  / 

0  / 

0  / 

25  09.6259 

27 

9-9573 

6 

9.6687 

33 

0-3313 

65  o 

35  o 

9.7586 

18  19.9134 

9 

9.8452 

27 

0.1548 

55  o 

10  .6286 

27 

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6 

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32 

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5° 

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18 

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9 

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27 

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50 

20 

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27 

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6 

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33 

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40 

20 

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18 

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9 

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27 

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4° 

3° 

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26 

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6 

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32 

•  3215 

30 

3° 

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9 

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26 

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6 

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20 

40 

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9 

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27 

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20 

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9 

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27 

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26  o 

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32 

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64  o 

36  o 

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18 

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26 

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54  ° 

10 

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26 

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6 

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IO 

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25 

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6 

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20 

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9 

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26 

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4° 

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26 

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6 

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30 

30 

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26 

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20 

40 

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50 

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26 

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3° 

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26 

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7 

.7226 

31 

•  2774 

10 

50 

.7877 

16 

.8975 

IO 

.8902 

26 

.1098 

IO 

28  o 

.6716 

24 

•9459 

6 

.7257 

30 

•  2743 

62  o 

38  o 

.7893 

I? 

.8965 

IO 

.8928 

26 

i 
.1072  52  o 

IO 
20 

.6740 
.6763 

23 
24 

•9453 
•9446 

7 
7 

.7287 
•7317 

3° 
3i 

•  2713 
.2683 

50 
40 

IO 

20 

.7910 
.7926 

16 
15 

.8955 
.8945 

IO 
10 

•  8954 
.8980 

26 
26 

.1046 

.1020 

50 
40 

3° 

.6787 

23 

•9439 

7 

.7348 

30 

.2652 

3° 

3° 

.7941 

16 

.8935 

10 

.9006 

26 

.0994 

30 

40 

.6810 

23 

•9432 

7 

•7378 

3° 

.2622 

20 

40 

•7957 

16 

.8925 

10 

.9032 

26 

.0968 

20  ' 

50 

•6833 

23 

•9425 

7 

.7408  30 

.2592 

IO 

5° 

•7973 

16 

.8915 

to 

.9058 

26 

.0942 

10 

29  o 

.6856 

22 

.9418 

7 

•  7438!  29 

.2562 

61  o 

39  ° 

.7989 

15 

.8905 

IO 

.9084 

26 

.0916  51  o 

IO 

.6878 

23 

.9411 

7 

•7467I  3° 

•  2533 

50 

IO 

.8004 

16 

.8895  ii 

.9110 

25 

.0890 

50 

20 

.6901 

22 

.9404 

7 

.7497  29 

•  2503 

40 

20 

.8020 

15 

^.8884  10 

•9*35 

26 

.0865 

40 

3° 

.6923 

23 

•9397 

7 

•7526)  30 

•  2474 

3° 

30 

•8035 

i5 

.8874  10 

.9161 

26 

.0839 

3° 

40 

.6946 

22 

•9390 

7 

.7556  29 

-.2444 

20 

40 

.8050 

16 

.8864  II 

.9187 

25 

.0813 

20 

5° 

.6968 

22 

•9383 

8 

•7585  29 

.2415 

IO 

50 

.8066 

15 

.8853  I0 

.9212 

26 

.0788 

10 

30  o 

.6990 

22 

•9375 

7 

-7614  30 

.2386 

00  0 

40  o 

.8081 

'  = 

.8843 

II 

.9238 

26 

.0762  '50  o 

IO 

.7012 

21 

.9368 

7 

.7644  29 

.2356 

50 

IO 

.8096 

15 

.8832 

1  ! 

.9264 

25 

.0736 

50 

20 

.7033 

22 

.9361 

8 

.7673  28 

.2327 

40 

20 

.8m 

H 

.8821 

1  1 

.9289 

26 

.0711 

40 

3° 

•  7°55 

21 

•9353 

7 

.7701  29 

.2299 

3° 

30 

.8125 

15 

.8810 

I'  ) 

•9315 

26 

.0685 

30 

40 

.7076 

21 

•9346 

8 

.7730  29 

.2270 

20 

40 

.8140 

15 

.8800 

II 

•9341 

25 

.0659 

20 

5° 

.7097 

21 

•9338 

7 

•7759 

29 

.2241 

IO 

5° 

•8i55 

14 

.8789 

I  I 

.9366 

26 

.0634 

IO 

31  o 

.7118 

21 

•9331 

8 

.7788 

28 

.2212 

59  *> 

41  o 

.8169 

15 

.8778 

1  1 

•9392 

25 

.060849  ° 

10 

.7139 

21 

•9323 

8 

.7816 

29 

.2184 

5° 

10 

.8184 

M 

.8767 

i  1 

.9417 

26 

.0583 

50 

20 

.7160 

21 

•93I5 

7 

.7845 

28 

•2155 

40 

20 

.8198 

15 

.8756 

:  I 

•9443 

25 

.0557 

40 

3° 

.7181 

2O 

.9308 

8 

.7873 

29 

-  .2127 

3° 

3° 

.8213 

14 

•8745 

!  2 

.9468 

26 

•  0532 

3° 

4° 
5° 

.7201 

S7222 

21 

20 

.9300 

.9292 

8 
8 

.7902 
•  793° 

28 

28 

.2098 
.2070 

20 

IO 

40 

So 

.8227 
.8241 

14 

14 

•8733!" 

.8722  II 

•9494 
•95J9 

25 
25 

.0506 

.0481 

20 

IO 

32  o 

.7242 

2O 

.9284 

8 

•7958 

28 

.2042 

58  o 

42  o 

•8255 

14 

.8711 

i  a 

•9544 

26 

.0456 

48  0 

IO 

.7262 

2O 

.9276 

8 

.7986 

28 

.2OI4 

50 

10 

.8269 

M 

.  8699 

1  1 

•957° 

25 

.0430 

5° 

20 

.7282 

20 

.9268 

:-, 

.8014 

28 

.1986 

40 

20 

.8283 

*4 

.8688,  12 

•9595 

26 

.0405 

4o 

3° 

.7302 

20 

.9260 

8 

.8042 

28 

.1958 

30 

30 

.8297 

14 

.8676  ii 

.9621 

25 

•  0379 

3° 

40 

.7322 

20 

.9252 

8 

.8070 

27 

.1930 

20 

40 

.8311 

*3 

.8665 

i  : 

.9646 

25 

•°354 

20 

So 

•7342 

19 

.9244 

8 

.8097 

28 

.1903 

IO 

50 

•  8324 

14 

.8653 

a 

.9671 

26 

.0329 

IO 

33  o 

.7361 

19 

.9236 

8 

.8125 

28 

•1875 

57  o 

43  o 

•8338 

*3 

.8641 

i  3 

.9697 

25 

.0303:47  o 

10 

.7380 

2O 

.9228 

9 

•  8153 

27 

.1847 

5° 

10 

•8351 

14 

.8629 

i  i 

.9722 

25 

.0278 

50 

20 

.7400 

19 

.9219 

8 

.8180 

28 

.1820 

40 

20 

.8365 

13 

.8618 

i  3 

•9747 

25 

-0253 

40 

3° 

.7419 

J9 

.9211 

8 

.8208 

27 

•1792 

30 

30 

.8378 

13 

.8606 

i  < 

.9772 

26 

.0228 

3° 

40 

.7438 

19 

.9203 

9 

.8235 

28 

•1765 

20 

4° 

.8391 

*4 

•8594 

1  ' 

•9798 

25 

.0202 

'  20 

50 

•7457 

19 

.9194 

8 

.8263 

27 

•J737 

IO 

50 

.8405 

13 

.8582 

13 

.9823 

25 

.0177 

IO 

34  o 

.7476 

18 

.9186 

9 

.8390 

27 

.1710 

56  o 

44  o 

.8418 

13 

.8569!  12 

.9848 

26 

.0152  46  o 

IO 

•7494 

19 

.9177 

8 

•8317 

27 

.1683 

50 

IO 

.8431 

*3 

•8557 

I  -j 

•9874 

25 

.OI26 

5° 

20 

-75*3 

18 

.9169 

9 

.8344 

27 

.1656 

4° 

20 

.8444 

*3 

•8545 

13 

.9899 

25 

.OIOI 

40 

3° 

•7531 

19 

.9160 

9 

•8371 

27 

.1629 

3° 

3° 

.8457 

12 

•8532 

ia 

.9924 

25 

.0076 

30 

40 

•7550 

18 

•9*5* 

9 

.8398 

27 

.1602 

20 

4° 

.8469 

*3 

.8520 

i  ^ 

•9949 

26 

.0051 

20 

5° 

.7568 

18 

.9142 

8 

.8425 

27 

•1575 

IO 

50 

.8482 

13 

.8507 

u: 

9-9975 

25 

.0025 

10 

35  o 

9.7586 

18 

9-9I34 

9 

9.8452 

27 

0.1548 

55  o 

45  o 

9.8495 

9.8495 

0.0000 

o.oooO|45  o 

Arc 

Cos. 

Df. 

Sin. 

Df. 

Cot.  Df. 

Tan. 

Arc 

Arc 

Cos.  1  Df. 

Sin. 

Df. 

Cot. 

Df. 

Tan. 

Arc 

68 


SQUARES    AND    SQUARE    ROOTS. 


XXIV.  —  Squares  and  Square  Roots. 

No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

i 

i 

1.  000 

5i 

2601 

7.141 

2 

4 

1.414 

52 

2704 

7.  211 

3 

9 

1.732 

53 

2809 

7.280 

4 

16 

2.000 

54 

2916 

7.348 

5 

25 

2.236 

55 

3025 

7.416 

6 

36 

2.449 

56 

3136 

7-483 

7 

49 

2.646 

57 

3249 

7.550 

8 

64 

2.828 

58 

3364 

7.616 

9 

Si 

3.000 

59 

348i 

7.681 

10 

100 

3.162 

60 

3600 

7.746 

ii 

121 

3.3T7 

61 

3721 

7.810 

12 

144 

3.464 

62 

3844 

7.874 

13 

169 

3.606 

63 

3969 

7-937 

14 

196 

3-742 

64 

4096 

8.000 

15 

225 

3.873 

65 

4225                  8.062 

16 

256 

4.000 

66 

4356 

8.124 

17 

289 

4.123 

67 

4489 

8.185 

18 

324 

4-243 

68 

4624                  8.246 

19 

361 

4.359 

69 

4761                  8.307 

20 

4OO 

4.472 

70 

4900                  8.367 

21 

441 

4o83 

71 

5041                  8.426 

22 

484 

4.690 

72 

5184                  8.485 

23 

529 

4.796 

73 

5329                  8.544 

24 

576 

4.899 

74 

5476                  8.602 

25 

625 

5.OOO 

75 

5625                  8.660 

26 

676 

5.099 

76 

5776                  8.718 

27 

729 

5.196 

77 

5929                  8.775 

28 

784 

5.292 

78 

6084                  8.832 

29 

84I 

5.385 

79 

6241                 8.888 

30 

900 

5.477 

80 

6400                 8.944 

31 

961 

5.568 

Si 

6561 

9.000 

32 

1024 

5.657 

82 

6724 

9-055 

33 

1089 

5-745 

83 

6889 

9.110 

34 

1156 

5.831 

84 

7056 

9.165 

35 

1225 

5.916 

85 

7225 

9.220 

36 

1296 

6.000 

86 

7396 

9.274 

37 

1369 

6.083 

87 

7569 

9.327 

33 

1444 

6.164 

88 

7744                 9.38r 

39 

1521 

6.245 

89 

7921                 9.434 

40 

I6OO 

6.325 

90 

Sioo                 9.487 

41 

1681 

6.403 

91 

8281                 9.539 

42 

1764 

6.481 

92 

8464 

9-592 

43 

1849 

6-557 

93. 

8649                 9.644 

44 

1936 

6.633 

94 

8836 

9.695 

45 

2025 

6.708 

95 

9025 

9-747 

46 

2116 

6.782 

96 

9216                 9.  798 

47 

2209 

6.856 

97 

9409                 9-849 

48 

2304 

6.928 

98 

9604                 9-899 

J9 

2401 

7.000 

99 

9801                 9-950 

50 

2500 

7.071 

IOO 

IOOOO                     IO.OOO 

SQUARES    AND    SQUARE    ROOTS.                                    69 

XXIV.  —  Squares  and  Square  Roots  —  Continued. 

No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

101 

I020I 

10.050 

151 

22801 

12.288 

102 

10404 

IO.  IOO 

152 

23104 

12.329 

103 

10609 

10.149 

153 

23409 

12.369 

104 

10816 

10.198 

154 

23716 

12.410 

105 

11025 

10.247 

155 

24025 

12.450 

I  O6 

11236 

10.296 

156 

24336 

12.490 

107 

11449. 

10.344 

157 

24649 

12.530 

108               11664 

10.392 

158 

24964 

12.570 

109               11881 

10.440 

159' 

25281 

12.610 

-    IIO                     I2IOO 

10.488 

160             25600 

12.650 

III                     I232I 

10.536 

161              25921 

12.689 

112                     12544 

10.583 

162 

26244 

12.728 

113                     12769 

10.630 

163 

26569 

12.767 

114                     12996                  10.677 

164 

26896 

12.806 

115                     13225 

10.724 

165              27225 

12.845 

116                13456 

10.771 

166              27556 

12.884 

117                13689              10.817 

167              27889 

12.923 

nS                13924              10.863 

168 

28224 

12.961 

119                14161              10.909 

169 

28561 

13.000 

120                     14400 

.10.954 

170     |         28900 

13.038 

121                     14641 

I  I  .  OOO 

171 

29241 

13.077 

122                     14884 

11.045 

172             29584 

I3.U5 

123                     15129 

1  1  .  09  1 

173 

29929 

I3.T53 

124                     15376 

II.  136 

174 

30276 

13.191 

125 

15625 

II.  I  80 

175 

30625 

13-229 

126 

15876 

11.225 

176 

30976 

13.266 

127 

16129 

11.269 

177 

31329 

13.304 

128 

16384 

11.314 

i/8 

31684 

13.342 

I29 

16641 

11.358 

179 

32041 

13-379 

130 

16900 

11.402 

1  80 

32400 

13-416 

131                     I7l6l 

11.446 

181 

32761 

13.454 

132                     17424 

11.489 

182 

33124 

I3.49I 

133                     17689 

11-533 

183 

33489 

13.528 

134                     17956 

11.576 

184 

33856 

13-565 

135                     18225 

11.619 

185 

34225 

13.601 

136                     18496 

11.662 

186 

34596 

13-638 

137                     18769 

11.705 

187 

34969 

13-675 

138                     19044 

n.747 

188 

35344                I3.7H 

139                     19321 

11.790 

189              35721                13.748 

140 

19600 

11.832 

190              36100                13.784 

141                     igSSi 

11.874 

191              36481                13.820 

142                     20164 

11.916 

192              36864                13-856 

143 

20449 

11.958 

193              37249                13-892 

144 

20736 

I  2  .  OOO 

194 

37636                13-928 

14-5 

21025 

I2.O42 

195 

38025              '13.964 

146                    2I3IO 

12.083 

196 

38416                14.000 

147                    2l6og 

12.  124 

197 

38809                14.036 

I48 

21904 

I2.I66 

198 

39204                14.071 

149                    222OI 

12.207 

199              39601                14.107 

150                    22500                  12.247                  200                  40000                     14.142 

7° 


SQUARES    AND    SQUARE    ROOTS. 


XXIV.  —  Squares  and  Square  Roots  —  Continued. 

No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

2OI 

40401 

14.177 

251 

63001 

15.843 

2O2 

40804 

14.213 

252 

63504 

15.875 

203 

41209 

14.248 

253 

64009 

15.906 

204 

41616 

14.283 

254 

64516 

15-937 

2O5 

42025 

14-318 

255 

65025 

15.969 

206 

42436 

14-353 

256 

65536 

16.000 

2O7 

42849 

14-387 

257 

66049 

16.031 

208 

43264 

14.422 

258 

66564 

16.062 

2O9 

43681 

14.457 

259 

67081 

16.093 

2IO        44IOO 

14.491 

260 

67600 

16.  125 

211         44521        14.526 

261     68121 

16.155 

212      44944      14-560 

262 

68644 

16.186 

213    45369    14.595 

263 

69169 

16.217 

214 

45796 

14.629 

264 

69696 

16.248 

215 

46225 

14.663 

265     70225 

16.279 

.    216 

46656 

14.697 

266 

70756 

16.310 

217 

47089 

14.731 

267 

71289 

16.340 

218 

47524 

14-765 

268 

71824 

16.371 

219 

47961 

14.799 

269 

72361 

16.401 

220 

48400 

14-832 

270 

72900 

16.432 

221 

48841 

14.866 

271 

73441 

16.462 

222 

49284 

14.900 

272 

73984 

16.492 

223 

49729 

14-933 

273 

74529 

16.523 

224 

50176 

14.967 

274  . 

75076 

16.553 

225 

50625 

i  5  .  ooo 

275 

75625 

16.583 

226 

51076 

15.033 

276 

76176 

16.613 

227 

51529 

15.067 

277 

76729 

16.643 

228 

51984 

15.100 

278 

77284 

16.673 

229 

52441 

I5-T33 

279 

77841 

16.703 

•    23O 

52900 

15.  166 

280 

78400 

16.733 

231 

5336i 

15.199 

281 

78961 

16.763 

232 

53824 

15-232 

282 

79524 

16.793 

233 

54289 

15.264 

283 

80089 

16.823 

234 

54756 

15-297 

284 

80656 

16.852 

235 

55225 

15.330 

285 

81225 

16.882 

236 

55696 

15-362 

286 

81796 

16.912 

237 

56169 

15.395 

287 

82369 

16.941 

238 

56644 

15.427 

288 

82944 

16.971 

239 

57121 

i  5  .  460 

289 

83521 

i  7  .  ooo 

240 

57600 

15-492 

290 

84100 

17.029 

241 

58081 

15.524 

291 

84681 

17.059 

242 

58564 

15.556 

292 

85264 

17.088 

243 

59049 

15.588 

293 

85849 

17-117 

244 

59536 

15.620 

294 

86436 

17.146 

245 

60025 

15.652 

295 

87025 

17.176 

246 

60516 

15-684 

296 

87616 

17-205 

247 

61009 

15.716 

297 

88209 

17.234 

248 

61504 

15.748 

298 

88804 

17-263 

249 

62001 

15.780 

299  i    89401      17.292 

250 

62500 

15.811 

300  j    90000      17.321 

SQUARES    AND    SQUARE    ROOTS. 

71 

XXIV.  —  Squares  and  Square  Roots  —  Continued. 

No. 

Square. 

Square  root. 

No'. 

Square. 

Square  root. 

» 

301 

90601 

17-349 

35i 

123201 

18.735 

302 

91204 

17.378 

352 

123904 

18.762 

303 

91809 

17.407 

353 

124609 

18,788 

304 

92416 

17.436 

354 

125316 

18.815 

305 

93025 

17.464 

355 

126025 

18.841 

306 

93636 

17.493 

356 

126736 

18.868 

307 

94249 

17.521 

357 

127449 

18.894 

308 

94864 

17.550 

358 

128164 

18.921 

309 

95481 

17.578 

359 

128881 

18.947 

310 

96100 

17.607 

360 

129600 

18.974 

311               96721             17-635 

361 

130321 

19.000 

312                Q7344 

17.664 

362 

131044 

19.026 

313 

97969 

17.692 

363 

131769 

I9.053 

3U                98596 

17.720 

364 

132496 

19.079 

315                99225 

17.748 

'365 

133225 

19.105 

316                99856 

17.776 

366 

133956 

19.131 

317               100489 

17.804 

367 

134689 

ig^s? 

318               101124 

17.833 

368 

135424 

19.183 

319              101761 

17.861 

369 

136161 

19.209 

320 

102400 

17.889 

370 

136900 

19-235 

321 

103041 

17.916 

37i 

137641 

19.261 

322 

103684 

17.944 

372 

138384 

19.287 

323 

104329              17.972 

373 

139129 

19-313 

324 

.    104976              18.000 

374 

139876 

19-339 

•        325 

105625 

18.028 

375 

140625 

19-365 

326 

106276              18.055 

376 

141376 

19.391 

327 

106929 

18.083 

377 

142129 

19.416 

328 

107584 

18.111 

378 

142884 

19.442 

329 

108241 

18.138 

379 

143641 

19.468 

330 

108900 

18.166 

380 

144400 

19.494 

331 

109561 

18.193 

38i 

145161 

19-519 

332 

110224 

18.221 

382 

145924 

19-545 

333 

110889 

18.248 

383 

146689 

19.570 

334 

111556              18.276 

384 

147456 

19.596 

335 

112225 

18.303 

385 

148225 

19.621 

336 

112896 

18.330 

386 

148996 

19.647 

337 

113569 

18.358 

387 

149769 

19.672 

338 

114244 

18.385 

388 

150544 

19.698 

339 

114921 

18.412 

389 

151321 

19.723 

340 

115600 

18.439 

390 

152100 

19.748 

34i 

116281 

18.466 

39i 

152881 

19-774 

342 

116964 

18.493 

392 

153664 

19.799 

343 

117649 

18.520 

393 

154449 

19.824 

344 

118336 

18.547 

394 

155236 

19.849 

345 

119025 

18.574 

395 

-  156025 

19-875 

346 

119716 

18.601 

396 

156816 

19.900 

347 

120409 

18.628 

397 

157609 

19.925 

348 

121104 

18.655 

398 

158404    • 

19.950 

349 

121801 

18.682 

399 

159201 

19.975 

350         %     122500             18.708 

400           160000 

;      20.000 

SQUARES    AND    SQUARE    ROOTS. 


XXIV. — Squares  and  Square  Roots — Continued. 


No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

401 

160801 

20.025 

45i 

203401 

21.237 

402 

161604 

20.050 

452 

204304 

21.260 

403 

162409 

20.075 

453 

205209 

21.284 

404 

163216 

20.100 

454 

206116 

•21.307 

405 

164025 

20.  125 

455 

207025 

21.331 

406 

164836 

20.  149 

456 

207936 

21.354 

407 

165649 

20.174 

457 

208849 

21.378 

408 

166464 

20.199 

458 

209764 

21.401 

409 

167281 

20.224 

459 

210681 

21.424 

410 

168100 

20.248 

460 

211600 

21.448 

411 

168921 

20.273 

461 

212521       21.471 

412 

169744      20.298 

462 

213-144 

21.494 

413 

170569      20.322 

463 

214369 

21.517 

414 

171396 

20.347 

464 

215296 

21.541 

415 

172225 

20.372 

465 

216225 

21.564 

416 

173056 

20.396 

466 

217156 

21.587 

417 

173889 

20.421 

467 

218089 

21.610 

418 

174724 

20.445 

468 

219024 

21.633 

419 

I7556I 

20.469 

469 

219961 

21.656 

420 

176400 

20.494 

47° 

220900 

21.679 

421 

177241 

20.518 

471 

221841 

21.703 

422 

178084 

20.543 

472 

222784 

21  .726 

423 

178929 

20.567 

473 

223729 

21.749 

424 

179776 

20.591 

474 

224676 

21.772 

425 

180625 

20.616 

475 

225625 

21.794 

426 

181476 

20.640 

476 

226576 

21.817 

427 

182329 

20.664 

477 

227529 

21.840 

428 

183184 

20.688 

4/8 

228484 

21.863 

429 

184041 

20.712 

479 

229441 

21.886 

430 

184900 

20.736 

480 

230400 

21.909 

43r 

185761 

20.761 

481 

231361 

21.932 

432 

186624 

20.785 

482 

232324 

21-954 

433 

187489 

20.809 

483 

233289 

21.977 

434 

188356 

20.833 

484 

234256 

22.000 

435 

189225 

20.857 

485 

235225 

22.023 

436 

190096 

20.881 

486 

236106 

22.045 

437 

190969 

20.905 

487 

237169 

22.068 

438 

191844 

20.928 

488 

238144 

22.091 

439 

192721 

20.952 

489 

239121 

22.113 

440 

193600 

20.976 

490 

240100 

22.130 

441 

194481 

2  1  .  OOO 

491 

241081 

22.159 

442 

195364 

21.024 

492 

242064 

22.I8I 

443 

196249 

21.048 

493 

243049 

22.2O4 

444 

197136      21.071 

494 

244036 

22.226 

445 

198025      21.095 

495 

245025 

22.249 

446 

198916      21.119 

496 

246016 

22.271 

447 

199809      21.142 

497 

247009 

22.293 

448 

200704      21.166 

498 

248004 

22.316 

449 

201601      21.190 

499 

249001 

22.338 

450 

202500      21.213 

500  1   250000      22.361 

SQUARES    AND    SQUARE    ROOTS. 


73 


XXIV. — Squares  and  Square  Roots — Continued. 


No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

SGI 

251001 

22.383 

55i 

303601 

23.473 

502 

252004 

22.405 

552 

304704 

23.495 

503 

253009 

22.428 

553 

305809 

23-516 

504 

254016 

22.450 

554 

306916 

23-537 

505 

255025 

22.472 

555 

308025 

23.558 

506 

256036 

22.494 

556 

309136 

23.580 

507 

257049 

22.517 

557 

310249 

23.601 

508 

258064 

22.539 

558 

311364 

23.622 

509 

259081 

22.561 

559 

312481 

23.643 

5io 

260100 

22.583 

560 

313600 

23.664 

BII 

261121 

22.605 

56i 

314721 

23-685 

512 

262144 

22.627 

562 

315844 

23.707 

5i3 

263169 

22.650 

563 

316969 

23.728 

5i4 

264196 

22.672 

564 

318096 

23-749 

515 

265225 

22.694 

.  565 

319225 

23.770 

5i6 

266256 

22.716 

566 

320356 

23.791 

5i7 

267289 

22.738 

567 

321489 

23.812 

5i8 

268324 

22.760 

568 

322624 

23-833 

519 

269361 

22.782 

569 

323761 

23-854 

520 

270400 

22.804 

570 

324900 

23.875 

521 

271441 

22.825 

57i 

326041 

23.  '896 

522 

272484 

22.847 

572 

327184 

23.917 

523 

273529 

22.869 

573 

328329 

23-937 

524 

274576 

22.891 

574 

329476 

23.958 

525 

275625 

22.013 

575 

330625 

23.979 

526 

276676 

22.935 

576 

331776 

24  .  ooo 

527 

277729 

22.956 

577 

332929 

24.021 

528 

278784 

22.978 

578 

334084 

24.042 

529 

279841 

23.000 

579 

335241 

24.062 

530 

280900 

23.022 

580 

336400 

24.083 

53i 

281961 

23-043 

5Si 

33756i 

24.104 

532 

283024 

23.065 

582 

338724 

24.125 

533 

284089 

23.087 

583 

339889 

24.145 

534 

285156 

23.  108 

584 

341056 

24.166 

535 

286225 

23.130 

585 

342225 

24.187 

536 

287296 

23.152 

586 

343396 

24.207 

537 

288369 

23.173 

587 

344569 

24.228 

538 

289444 

23.195 

588 

345744 

24.249 

539 

290521 

23.216 

589 

346921 

24.269 

540 

291600 

23.238 

590 

348100  , 

24.290 

54i 

292681 

23-259 

59i 

349281 

24.310 

542 

293764 

.  23.281 

592 

350464 

24.331 

543 

294849 

23.302 

593 

351649 

24.352 

544 

295936 

23.324 

594 

352836 

24.372 

545 

297025 

23-345 

595 

354025 

24.393 

546 

298116 

23.367 

596 

355216 

24.413 

547 

299209 

23.388 

597 

356409 

24.434 

548 

300304 

23.409 

598 

357604 

24.454 

549 

301401 

23.431 

599 

358801 

24.474 

550 

302500 

23.452 

600 

360000 

24.495 

74 


SQUARES   AND    SQUARE    ROOTS. 


XXIV. — Squares  and  Square  Roots — Continued. 


No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

601 

361201 

24.515 

651 

423801 

25.5T5 

602 

362404 

24-536 

652 

425104 

25.534 

603 

363609 

24.556 

653 

426409 

25.554 

604 

364816 

24-576 

654 

427716 

25.573 

605 

366025 

24-597 

655 

429025 

25.593 

606 

367236 

24.617 

656 

430336 

25.612 

607 

368449 

24-637 

657 

431649 

25.632 

608 

369664 

24-658 

658 

432964 

25.652 

609 

370881 

24.678 

659 

434281 

25-671 

610 

372100 

24.698 

660 

4356oo 

25.690 

611 

373321 

24.718 

661 

436921 

25.710 

612 

374^44 

24.739 

662 

438244 

25.720 

613 

375769 

24.759 

663 

439569 

25.749 

614 

376996 

24.779 

664 

440896 

25.768 

615 

378225 

24.799 

665 

442225 

25.788 

616 

379456 

24.819 

666     443556 

25.807 

617 

380689 

24-839 

667     444889 

25.826 

618 

381924 

24.860 

668 

446224 

25.846 

619 

383161 

24.880 

669 

447561       25.865 

620 

384400 

24  .  900 

670 

448900 

25.884 

621 

385641 

24.920 

671 

450241 

25.904 

622 

386884 

24  .  940 

672 

451584 

25.923 

623 

388129 

24.960 

673 

452929 

25.942 

624 

389376 

24.980 

674 

454276 

25.962 

625 

390625 

25.000 

675 

455625 

25.981 

626 

391876 

25.020 

676 

456976 

26.000 

627 

393129 

25.040 

677 

458329 

26.019 

628 

394384 

25.060 

678 

459684 

26.038 

629 

395641 

25.080 

679 

461041 

26.058 

630 

396900 

25.100 

680 

462400 

26.077 

631 

398161 

25.120 

681 

463761 

26.096 

632 

399424 

25.140 

682 

465124 

26.115 

633 

400689 

25.160 

683 

•466489 

26.134 

634 

401956 

25.180 

684 

467856 

26.153 

635 

403225 

25.200 

685 

469225 

26.173 

636 

404496 

25.220 

686 

470596 

26.192 

637 

405769 

25.239 

687     471969 

26.211 

638 

407044 

25-259 

688     473344 

26.230 

639 

408321 

25.278 

689     474721 

26.249 

640 

409600 

25.298 

690 

476100 

26.268 

641 

410881 

25.318 

691 

477481 

26.287 

642 

412164 

25.338 

692     478864 

26.306 

643 

413449 

25-357 

693    '  480249 

26.325 

644 

414736 

25.377 

694     481636 

26.344 

645 

416025 

25.397 

695     483025      26.363 

646 

417316 

25-417 

696     484416      26.382 

647 

418609 

25.436 

697     485809      26.401 

648 

419904 

25.456 

698     487204      26.420 

649 

421201 

25-475 

699     488601 

26.439 

650 

422500 

25.495 

700  '   490000 

26.458 

SQUARES    AND    SQUARE    ROOTS. 


75 


XXIV. — Squares  and  Square  Roots — Continued. 


No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

701 

491401 

26.476 

751 

564001 

27.404 

702 

492804 

26.495 

752 

565504 

27-423 

703 

494209 

26.514 

753 

567009 

27.441 

704 

495616 

26.532 

754 

568516 

27-459 

705 

497025 

26.552 

755 

5/0025 

27-477 

706 

498436 

26.571 

756 

571536 

27.495 

707 

499849 

26.589 

757 

573049 

27.514 

708 

501264 

26.608 

758 

574564 

27.532 

709 

502681 

26.627 

759  ! 

.  576081 

27.550 

710 

504100 

26.646 

760  | 

577600 

27.568 

711 

505521 

26.665 

761 

579121 

27.586 

712 

506944 

26.683 

762 

580644. 

27.604 

713 

508369 

26.702 

763 

582169 

27.622 

7J4 

509796 

26.721 

764 

583696 

27.641 

715 

511225 

26.739 

765 

585225 

27.659 

716 

512656 

26.758 

766 

586756 

27.677 

7J7 

514089 

26.777 

767 

588289 

27.695 

718 

515524 

26.796 

768 

289824 

27.713 

719 

516961 

26.814 

769 

591361 

27.731 

720 

518400 

26.833 

770 

592900 

27.749 

721 

519841 

26.851 

77i 

594441 

27.767 

722 

521284 

26.870 

772 

595984 

27-785 

723 

522729 

26.889 

773 

597529 

27.803 

724 

524176 

26.907 

774 

599076 

27.821 

725 

525625 

26.926 

775 

600625 

27-839 

726 

527076 

26.944 

776 

602176 

27.857 

727 

528529 

26.963 

777 

603729 

27.875 

728 

529984 

26.981 

778 

605284 

27.893 

729 

53I44I 

27.000 

779 

606841 

27.911 

730 

532900 

27.019 

780 

608400 

27.928 

731 

534361 

27.037 

781 

609961 

27.946 

732 

535824 

27-055 

782 

611524 

27.964 

733 

537289 

27-074 

783 

613089 

27.982 

734 

538756 

27.092 

784 

614656 

28.000 

735 

540225 

27.  in 

785 

616225 

28.018 

736 

541696 

27.129 

786 

617796 

28.036 

737 

543169 

27.148 

787 

619369 

28.054 

738 

544644 

27.166 

788 

620944 

28.071 

739 

546121 

27.185 

789 

622521 

28.089 

740 

547600 

27.203 

790 

624100 

28.107 

741 

549081 

27.221 

791 

625681 

28.125 

742 

550564 

27.240 

792 

627264 

28.142 

743 

552049 

27.258 

793 

628849 

28.160 

744 

553536 

27.276 

794 

630436 

28.178 

"745 

555025 

27.295 

795 

632025 

28.196 

746 

556516 

'  27.313 

796 

633616 

28.213 

747 

558009 

27.331 

797 

635209 

28.231 

748 

559504 

27.350 

798 

636804 

28.249 

749 

561001 

27.368 

799 

638401 

28.267 

750 

562500 

27.386 

800 

640000 

28.284 

76 

SQUARES  AND  SQUARE  ROOTS. 

XXIV.  —  Squares  and  Square  Roots  —  Continued. 

No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

Soi 

641601 

28.302 

851 

724201 

29.172 

802 

643204 

28.320 

852 

725904 

29.  189 

803 

644809      28.337 

853 

727609 

29.206 

804 

646416      28.555 

854 

729316 

29.223 

805 

648025      28.373 

855 

731025 

29.240 

806 

649636 

28.390 

856 

732736 

29.257 

807 

651249      28.408 

857 

734449 

29.275 

808 

652864      28.425 

858 

736164 

29.292 

809 

654481 

28.443 

859 

737881 

29.309 

8  10 

656100 

28.460 

860 

739600 

29.326 

Sn 

657721 

28.478 

861 

741321 

29-343 

812 

659344 

28.496 

862 

743044 

29.360 

813 

660969 

28.513 

863 

744769 

29-377 

814 

662596   |   28.531 

864 

746496 

29.394 

815 

664225 

28.548 

865 

748225 

29.411 

Si6 

665856 

28.566 

866 

749956 

29.428 

817 

667489 

28.583 

867 

751689 

29.445 

818 

669124 

28.601 

868 

753424 

29.462 

8ig 

670761 

28.618 

869 

755i6i 

29.479 

820 

672400 

28.636 

870 

756900 

29.496 

821 

674041      28.653 

871 

758641 

29-513 

822 

675684 

28.671 

872 

760384 

29-530 

823 

677329 

28.688 

873 

762129 

29.547 

824 

678976 

28.705 

874 

763876 

29-563 

825 

680625 

.  28.723 

875 

765625 

'29.580 

826 

682276 

28.740 

876 

767376 

29.597 

827 

683929 

28.758 

877 

769129 

29.614 

828 

685584 

'  28.775 

878 

770884 

29.631 

829 

687241 

28.792 

879 

772641 

29.648 

830 

688900 

28.810 

SSo 

774400 

29.665 

S3i 

690561 

28.827 

881 

776161 

29.682 

832 

692224 

28.844 

882 

777924 

29.698 

833 

693889 

28.862 

883 

779689 

29-7I5 

834 

695556 

28.879 

884 

781456 

29.732 

835 

697225 

28.896 

885 

783225 

29.749 

836 

698896 

28.914 

886 

784996 

29.766 

837 

700569 

28.931 

887 

786769 

29.783 

838 

702244      28.948 

888 

788544 

29.799 

839 

703921 

28.965 

889 

790321 

29.816 

840 

705600 

28.983 

890 

792100 

29-833 

841 

707281      29.000 

891 

793881 

29.850 

842 

708964      29.017 

892 

795664 

29.866 

843 

710649 

29.034 

893 

797449 

29.883 

844 

712336      29.052 

894 

799236 

29.900 

845 

714025      29.069 

895 

801025 

29.917 

846 

715716      29.086 

896 

802816 

29.933 

847 

717409 

29.103 

897 

804609 

29.950 

848 

719104      29.120 

898 

806404 

29.967 

849 

720801      29.138 

899 

808201 

29.983 

850 

722500      29.155      goo     810000       30.000 

SQUARES  «AND    SQUARE    ROOTS. 


77 


XXIV. — Squares  and  Square  Roots — Continued. 


No. 

Square. 

Square  root. 

No. 

Square. 

Square  root. 

901 

811801 

30.017 

95i 

904401 

30.838 

902 

813604 

30-033 

952 

906304 

30.854 

903 

815409 

30.050 

953 

908209 

30.871 

904 

817216 

30.067 

954 

910116 

30.887 

905 

819025 

30.083 

955 

912025 

30.903 

906 

820836 

30.  TOO 

956 

9^936 

30.919 

907 

822649 

30.116 

957     915849 

30.935 

908 

824464 

30.133 

958 

917764 

30.952 

909 

826281 

30.150 

959 

919681 

30.968 

910 

828100 

30.166 

960 

921600 

30.984 

911 

829921 

30.183 

961 

923521 

3  i  .  ooo 

912 

331744 

30.199 

962 

925444 

31.016 

913 

833569 

30.216 

963 

927369 

31.032 

914 

835396 

30.232 

964 

929296 

31.048 

9J5 

837225 

30.249 

965 

931225 

31.064 

916 

839056 

30.265 

966 

933156 

31.081 

917 

840889 

30.282 

967 

935089 

31-097 

918 

842724 

30.299 

968 

937024 

3i.ii3 

919 

844561 

30.315 

969 

938961 

31.129 

920 

846400 

30.332 

970 

940900 

3I.I45 

921 

848241 

30.348 

971 

942841 

31.161 

922 

850084 

30.364 

972 

944784 

3LI77 

923 

851929 

30.381 

973 

946729 

3LI93 

924 

853776 

30.397 

974 

948676 

3  i  •  209 

925 

855625 

30.414 

975 

950625 

31-225 

tL 
926 

857476 

30.430 

976 

952576 

31.241 

927 

859329 

30.447 

977 

954529 

3L257 

928 

861184 

30.463 

978 

956484 

3L273 

929 

863041 

30.480 

979 

958441 

31.289 

930 

864900 

30.496 

980 

960400 

31.305 

931 

866761 

30.512 

981 

962361 

31-321 

932 

868624 

30.529 

982 

964324 

31-337 

933 

870489 

30.545 

983 

966289 

31-353 

934 

872356 

30.561 

984 

968256 

31-369 

935 

-j  *j  +j 

874225 

30.578 

985 

970225 

3L385 

936 

876096 

30.594 

986 

972196 

31.401 

937 

877969 

30.610 

987 

974169 

3I.4I7 

93S 

879844 

30.627 

988 

976144 

3L432 

939 

881721 

30.643 

989 

978121 

31.448 

940 

883600 

30.659 

990 

980100 

31.464 

941      885481 

30.676 

991 

982081 

31.480 

942 

887364 

30.692 

992 

984064 

31.496 

943 

889249 

30.708 

993 

986049 

31.512 

944 

891136 

30.725 

994 

988036 

31-528 

945 

893025 

30.741 

995 

990025 

3L544 

946 

894916 

30.757 

996 

992016 

31-560 

947 

896809 

30.773 

997 

994009 

31-575 

948 

898704 

30.790 

998 

996004 

3I.59I 

949 

900601 

30.806 

999 

998001 

31.607 

950 

902500 

30.822 

IOOO 

I  000000 

31.623 

78               INTERPOLATION  BY  DIFFERENCES. 

BessePs  Co-efficients. 

f.i 

2d  diff. 

3d  diff. 

4th  diff. 

!i 

2d  diff. 

3d  diff. 

4th  diff. 

o  ." 

o  £ 

^ 

f  '~~T 

t  t~l  *~l 

t   *~2 

t/)  ° 

/  '-1 

tJ-i  t-\ 

f_'-* 

ii 

2 

2   3 

4 

11 

2 

2   3 

4 

O.  OI 

—  .00495 

.00081 

.  00083 

0.51 

-•  12495 

—  .  00042 

•  02343 

.  02 

.  00980 

.00157 

.  00165 

•  52 

.  12480 

.  00083 

.  02340 

.03 

•01455 

.  00228 

.  00246 

•53 

•  12455 

.00125 

•  02334 

.04 

.  OI92O 

.  00294 

.  00326 

•54 

.  I242O 

.  00166 

.02327 

.05 

•02375 

.  00356 

.  00405 

•55 

•  12375 

.  00206 

.02318 

.06 

.  O282O 

.  00414 

.  00483 

•56 

.  12320 

.  00246 

.  02306 

.07 

•  03255 

.  00467 

.  00560 

•57 

.12255 

.  00286 

.  02293 

.08 

.03680 

.00515 

.  00636 

.58 

.  I2l8o 

.  00325 

.  02278 

.09 

•  04095 

.  00560 

.  00711 

•59 

.  12095 

.  00363 

.  02260 

.  10 

.  O45OO 

•  .  00600 

.  00784 

.  60 

.  I20OO 

.  00400 

.  02240 

.  II 

.04895 

.  00636 

.  00856 

.61 

.11895 

.  00436 

.  02218 

.  12 

.  05280 

.  00669 

.  00927 

.62 

.11780 

.  00471 

.  02194 

•13 

•  05055 

.  00697 

.  00996 

•63 

.11655 

.  00505 

.02169 

.  14 

.  O6O2O 

.  00723 

.  01064 

.64 

.  U520 

.  00538 

.  02141 

•  J5 

•06375 

•  00744 

.01130 

.65 

."375 

.  00569 

.  O2III 

.  16 

.  06720 

.  00762 

.01195 

.66 

.  II22O 

.  00598 

.  O2o8o 

.17 

•  07055 

.  00776 

.01259 

.67 

.11055 

.  00626 

.  02046 

.18 

.07380 

.  00787 

.01321 

.68 

.  10880 

.  00653 

.  O2OIO 

.19 

.  07695 

.  00795 

.01381 

.69 

.  10695 

.00677 

.01973 

.20 

.  08000 

.  00800 

.  01440 

.70 

.  10500 

.  00700 

.01934 

.21 

.  08295 

.  00802    .  01497 

•71 

.  10295 

.00721 

.01893 

.  22 

.  08580 

.00801    .01553 

.72 

.  IOO8O 

.  00739 

.01850 

•23 

.08855 

.  00797 

.01606 

•  73 

.09855 

.  00756 

.  01805 

.24 

.  O9I2O 

.  00790 

.01658 

•  74 

.  09620 

.  00770 

.01758 

•25 

•09375 

.  00781 

.01709 

•  75 

•09375 

.  00781 

.01709 

.26 

.  09620 

.  00770 

.01758 

•  76 

.09120 

.  00790 

.01658 

•27 
.28 

•  09855 
.  10080 

.  00756 
•  00739 

.01805 
.  01850 

•  77 
•  78 

.08855 
.  08580 

.  00797 

.  00801 

.  Ol6o6 

•OI553 

.29 

.  10295 

.00721 

.01893 

•  79 

.  08295 

.  00802 

.01497 

•30 

.  10500 

.  00700 

.01934 

.80 

.  O8OOO 

.  00800 

.  01440 

•31 

.  10695 

.  00677 

•01973 

.81 

.  07695 

.  00795 

.  01381 

•32 

.  10880 

.  00653 

.  O2OIO 

.82 

.  07380 

.  00787 

.01321 

•33 

•II055 

.  00626 

.  02046 

•83 

.07055 

.  00776 

.01259 

•34 

.  II22O 

.  00598 

.  02080 

•  84 

.  06720 

.  00762 

.01195 

•35 

.H375 

.  00569 

.  O2III 

.85 

•06375 

.  00744 

.  01130 

•36 

.  H520 

.  00538 

.02141 

.86 

.  O6O2O 

.  00723 

.  01064 

•37 

."655 

.  00505 

.02169 

.87 

•  05655 

.  00697 

.  00996 

•38 

.  II78O 

.  00471 

.02194 

.88 

.  05280 

.  00669 

.  00927 

•39 

.11895 

.  00436 

.  O22l8 

.89 

•  04895 

.  00636 

.  00856 

.40 

.  12000 

.  00400 

.  O224O 

.90 

.  04500 

.  00600 

.  00784 

•  4i 

•  12095 

.  00363 

.  O226o 

.91 

•  04095 

.  00560 

.00711 

•  42 

.  I2I80 

.  00325 

.02278 

•  92 

.  03680 

.00515 

.  00636 

•43 

•  12255 

.  00286 

.  02293 

•93 

•  03255 

.  00467 

.  00560 

•  44 

.  12320 

.  00246 

.  02306 

.94 

.  O282O 

.  00414 

.  00483 

•45 

•  12375 

.  00206 

.02318 

•  95 

•02375 

.  00356 

.  00405 

.46 

.  12420 

.  00166 

.  02327 

.96 

.  OI92O 

.  00294 

.00326 

•  47 

•  12455 

.  00125 

•  02334 

•  97 

.01455 

.  00228, 

.  00246 

.48 

.  12480 

.  00083 

.  02340 

.98 

.  00980 

.00157" 

.  00165 

.49 

.  12495 

.  00042 

•  02343 

•  99 

•  00495 

.  00081 

.  00083 

•50 

-.  I2500 

.  00000 

.02344 

I.  00 

—  .00000 

—  .  ooooo 

.  ooooo 

1 

TABLES  AND   FORMULAE. 


PART    II. 


GEODESY. 


GEODESY. 


XXV.— Reduction  to  Center  of  Station. 
Call— 

P  the  place  of  the  instrument ; 
C  the  center  of  the  station ; 

O  the  angle  at  P,  between  two  objects,  A  and  B ; 
y  the  angle  at  P,  between  C  and  the  /(//-hand  object,  B  ; 
r  the  distance,  C  P ; 
C  the  unknown  angle  at  C ; 

D*the  distance  A  C ;  and  • 

G  the  distance  B  C, 
then — 

c  =  o  4- ;  sin  IP  "i"  ~v)  -     r^_y 

D  sin  i"  Gsin'i" 

In  the  use  of  this  formula  proper  attention  should  be  paid  to 
the  signs  of  sin  (O  +J')  and  sin_y;  for  the  first  term  will  \>Q posi 
tive  when  (O  +j};)  is  less  than  180°,  (the  reverse  with  sin  y})  D 
being  the  distance  of  the  /7g///-hand  object,  the  graduation  of  the 
instrument  running  from  left  to  right. 

r  being  small,  the  lengths  of  D  and  G  are  computed  with  the 
angle  O. 

XXVI. — Reduction  to  Center  of  Signal  Observed,  or  Correction  for 
Phase  in  Tin  Cones  Used  as  Signals. 

,    r  cos2  i  Z 

Correctio    =  i  -       .    ^ 
D  sin  i" 

where — 

r  =  radius  of  the  signal ; 

Z  =  angle  at  the  point  of  observation  between  the  sun  and 

the  signal ;  and 
D  =  the  distance. 


82  GEODESY. 


XX  VI  I.—  Spherical  Excess. 

^  _         S  a  b  sin  C 

~  pTirn"  ~  :  T7-2  sin  i" 

S  being  the  area  of  the  triangle  ;  r,  the  radius  of  the  earth. 
a  b  sin  c 


=  V  s  (s-a)  (7^7(7^7) 

tf    _|_    £   _j_    f 

j  bemer  =  —  !  -  •  — 

2 

Between  latitudes  45°  and  25°  the  spherical  excess  amounts  to 
about  i"  for  an  area  of  75.5  square  miles. 

Hence,  if  the  area  in  square  miles  be  known,  a  close  approxi 
mation  to  the  spherical  excess  will  be  had  by  dividing  the  area 
by  75-5- 

log  mean  radius  of  the  ear.h  in  yards  =  6.8427917 

If  the  three  angles  of  a  triangle  are  assumed  to  have  been 
equally  well  determined,  the  previous  determination  of  the  spher 
ical  excess  is  not  necessary  for.  the.  calculation  of  the  sides,  though 
it  will  be  required  for  estimating  the  relative  accuracy  of  the  ob 
servations;  for  the  sides  of  a  spherical  triangle  may  be  com 
puted  as  if  they  were  rectilineal  when  one-third  the  excess  of  the 
sum  of  .the  three  angles  above  180°  is  deducted  from  each  of  the 
three  observed  angles.  Then  — 

side  b  —  side  a  sin  (B  —  J  E)  4-  sin  (A  —  J  E) 

For  large  triangles  : 

a  b  sin  C  (i  +  e2  cos  2  L) 
2  A2  sin   i" 

A  being  the  equatorial  radius,  and  L  the  mean  latitude  of  the 
three  stations. 


GEODESY.  83 


XXVIII. — To  Reduce  the  Length  of  an  Inclined  Base  to  Horizontal 

Measure. 

Let— 

B  be  the  length  of  the  base  on  the  inclined  plane ; 
b  that  reduced  to  the  horizontal  plane  ;   and 
0  the  inclination, 

then — 

b  =  B  cos  O 

But  as  0  is  generally  a  small  angle,  and  need  not  be  known  with 
extreme  precision,  it  is  better  to  compute  the  excess  of  B  above 
b ;  and,  supposing  0  to  be  given  in  minutes, 

B— b  =  B  ( 1  -cos  0)  =  2  B  sin2  °-  =  £  B  O2  sin2  I/  =  ^l1'  tf  B 

2  2 

or,    B  — />  =  0.00000004231  O2  B 
or,  by  logarithms, 

log  (B  —  b)  =  const  log  2.626422  -j-  2  log  0  -f  log  B 

XXIX. —  To  Reduce  a  Broken  Base  to  a  Straight  Line. 

Let- 

a  and  b  be  the  given  sides ;  and 
C  the  contained  angle,  very  nearly  180°. 
Make  C  =  180°  —  0;  0  being  small,  and  cos  0  —  r  —  £  O2, 
then — 


=  c7  -j-  ^  -  O.OOOOOOO42^I     X    -- 


being  expressed  in  minutes. 

log  0.00000004231  =  2.626422 


84  GEODESY. 


XXX. — To  Find  the  Length,  B  D  =  x,  of  a  Portion  of  a  Straight 
Line,  A  H,  Knowing  the  Two  Other  Portions,  A  B  =  a,  D  H  =  b, 
and  also  the  Angles  a,  ft,  y,from  any  Exterior  Station,  C,  between 
B  and  A,  D  and  A,  and  H  and  A. 

The  problem  being  intended  to  supply  by  observation  any  por 
tion  of  a  base  which  cannot  be  directly  measured — 

tan2  y  =  -4-^A  x  sin  P  sin  (r— «) 
(a — i>f     sin  a  sin  (Y — ft) 

a  -j-  b  ,    a  —  b 

2  2   COS  tp 


XXXI.  —  To  Reduce  a  Measured  Base  to  the  Level  of  the  Sea. 

Let— 

r  represent  the  radius  of  the  earth  (or  better,  the  normal,  N,) 

corresponding  to  the  base  b  at  the  level  of  the  sea  ;  and 
r  4-  a  the  radius  referred  to  the  level  of  the  measured  base  B, 
then  — 


r+a  :  r\\  B  :  ^  ==  B  x  —  - 

r+a 


and 


But  the  radius  of  the  earth  being  very  great  in  comparison  to 
the  difference  of  level,  a,  we  have  the  correction  8  sufficiently  ac 
curate  by  retaining  only  the  first  term  ;  hence  — 


GEODESY.  85 


XXXII.  —  Correction  for  Temperature  in  Metallic  Rods. 

Let— 

e  —the  linear  expansion  for  i°  of  Fahrenheit; 
/  =  the  length  of  the  rod  before  expansion  ; 
/'  =  the  length  of  the  rod  after  expansion  ; 
/  =  the  number  of  degrees  Fahrenheit, 

then- 

total  expansion  =  e  t 

and  — 


The  following  expansions  were  adopted  by  Mr.  Hassler  in  his 
comparisons  of  weights  and  measures,  (report  of  1832  :) 

Expansion  for  i°  F.  =  c  For  I    in  a  yard's  length. 

Platinum  .  .0.0000051344.  ..  .0.0001848384  English  inches. 
Brass  bar.  .0.00001050903.  .  .0.00037832508 
Iron  bar.  .  .0.000006963535  .  .0.000240687260 

Other  authorities  : 

Expansion  for  1°  Y.—c  ''/or  I    in  a  yard's  length. 

Brass  bar.  .0.000010480  .....  0.0003772800  Eng.  in.  Bailey. 
Brass  rod.  .0.0000105155  ...  .0.0003785580       "        Roy. 
Brass  rod.  .0.0000106666.  ..  .0.0003839976       "        Troughton. 
Brass  wire  .  .  0.0000107407  ....  0.0003866652       "         Smeaton. 
Iron  bar.  .  .0.0000069907  ...  .0.0002516652       "         Smeaton. 
Steel  rod.  .  .0.0000063596.  ..  .0.0002289456       "         Roy. 

Glass  barom 
eter-tubes  .0.0000043119  ...  .0.0001552284      "          Roy. 

White     Nor 
way  pine  .  .0.0000022685  ...  .0.0000816660       "         Kater. 


86 


GEODESY. 


XXXIII. — Measurement  of  Distances  by  Sound. 

The  velocity  of  sound,  in  one  second  of  time,  at  32°  Fahrenheit, 
is  about  1090  English  feet.  For  any  higher  temperature, 

7>=io89ft.42  V  i  +  (/  —  32°)  x  0.00208" 
/  being  the  temperature  in  degrees  Fahrenheit. 

The  velocity  of  sound  through  the  air  is  independent  of  the 
barometric  pressure,  and  experiments  show  it  to  be  sensibly  un 
affected  by  its  hygrometrical  state  of  moisture  and  dryness ;  by 
the  nature  of  the  sound  itself,  whether  produced  by  a  blow,  gun 
shot,  the  voice,  or  a  musical  instrument ;  by  the  original  direction 
of  the  sound,  whether,  for  instance,  the  muzzle  of  a  gun  is  turned 
one  way  or  the  other ;  or. by  the  nature  and  position  of  the  ground 
over  which  the  sound  is  conveyed. 

It  is  affected  by  the  wind ;  but,  in  ordinary  cases,  likely  to  be 
selected  for  experiment,  its  influence  would  be  almost  inappre 
ciable. 


Velocity  and  Force  of  the    Wind. 


•  Velocity  in  — 

Pressure  on 
I  square  foot. 

Common  designations  of  the  force  of  the 
winds. 

I  hour. 

i  second. 

, 

Miles. 

Feet. 

Pounds. 

i 

1.47 

0.005 

Hardly  perceptible. 

2 

3 

2-93 
4.40 

O.O2O 

0.044 

Just  perceptible.  • 

4 

5 

5-87 
7-33 

0.079 
0.123 

Gentle,  pleasant  wind. 

10 
15 

14.67 

22.0O 

0.492 
1.107 

Pleasant,  brisk  breeze. 

20 

25 

29-34 
36.67 

1.968 
3-075 

Very  brisk. 

30 

44.01                     4.429 

Pli^h  wind. 

35 

51.34                    6.027 

40 
45 

58.68 

66.01 

7-873 
9-963 

Very  high. 

5o 

73-35             12.300 

A  storm  or  tempest. 

60 

88.02 

I7-7I5 

A  great  storm. 

80 

117.36             3I-490 

A  hurricane. 

IOO 

146.70 

49.200 

A  hurricane  that  tears  up  trees,   carries 

buildings  before  it,  &c. 

GEODESY.  87 


XXXIV.  —  For  Reconnaissances. 

"THREE-POINT  PROBLEM." 

At  a  point,  P,  from  whence  are  to  be  seen  three  points,  A,  C, 
B,  forming  a  triangle,  the  elements  (/.  <?.,  the  angles  and  sides)  of 
which  are  known,  measure  the  angles  A  P  C  and  C  P  B;  then, 
required  to  determine  the  direction  and  distance  of  the  point  P 
from  each  object. 

Make— 

A  C=<z; 
B  C  =  J; 

B  C  A  =  C  ; 
A  P  C  =  P  ;    and 
C  P  B  =  P'  ; 
also,  make  — 

R  =  360°  -  P  -  P'  -  C  ; 
x=C  A   P;, 
}>  =  P  B  C. 
Then  will  — 


. 

sm  P  cos  R 

—  R  _  x 


J 


The  use  of  these  formulae  need  not  be  embarrassing  if  care  is 
taken  in  properly  applying  the  signs  of  cos  R  and  cot  R.  When 
R  is  less  than  90°  both  cos  R  and  cot  R  are  plus;  between  90°  and 
i8o°both  are  minus;  between  i8o°and  270°  cos  R  is  minus  and 
cot  R  plus;  between  270°  and  360°  cos  R  is  plus  and  cot  R 
minus. 

This  problem  is  indeterminate  when  P  falls  upon  the  circumfer 
ence  of  the  circle  passing*  through  A  B  C.  A  case  of  this  nature 
is  of  rare  occurrence,  however,  in  practice. 

For  the  more  general  form  of  this  problem,  where  the  angles 
are  measured  from  the  point  sought  to  any  number  of  given 
points,  to  fix  its  position,  see  Coast  Survey  Report  of  1864. 


38  GEODESY. 


XXXV.  —  For  Computing  the  Principal  Geodetic  Quantities  De 
pending  on  the  Spheroidal  Figure  of  the  Earth  at  any  Green. 
Latitude. 


Eccentricity  of  the  Earth  =  e  =    - 


Ellipticity  =  E  =-  =  i  —  - 


or,  very  nearly  —  : 


Normal  ending  at  minor  axis  (or  radius  of  curvature  of  a  section 
perpendicular  to  the  meridian)  =  N 

a 

=  (T^~sir?T)"i 
Normal  ending  at  major  axis  .....  =  N'  =  N  (i  —  e2} 


Tangent  ending  at  minor  axis.  .  .  =  /     =  N  cot  L 
Tangent  ending  at  major  axis.  .  .  .  =  T  =  N  tan  L  (i  —  e2} 
Radius  of  the  parallel  ..........  =  />    =  N  cos  L 

N3 
Radius  of  curvature  of  the  meridian  =  R  =  —  (i  —  e2) 

=      a  (i--*2} 
(i  —  <f2sin2  L)3 

Radius  of  curvature  of  a  section  making  an  angle  7.  with  the 


meridian 


~  N 
Radius   of   the   earth  ..........  =  ;- 


Equatorial  radius 
Polar  radius 
The  given  latitude 


GEODESY. 


XXXVI. — Numerical  Values  of  Some  of  the  Preceding  Quanti 
ties,  from  a  Discussion  by  BESSEL  in  the  "  Astronomische  Nach- 

richtcn"  No.  438. 

a  =  equatorial  radius  =3272077.14  toises 

log  =  6.5148235337 
b  =  polar  radius  =  3261 139.33  toises 
log  =  6. 5133693539 

Ratio  of  the  toise  to  the  metre,  law  of  France,  December  10, 
1799: 

T  =  i1". 9490363;  'log  =  0.2898 1 99300 

whence  in  metres — 

rt=6377397m.i5;  log  =  6.8046434637 

b  =  6356078"'. 96  ;  log  =  6.8031892839 

Ratio  of  the  axes : 

a  :  b  : :  299.1528  :  298.1528 
Mean  uncertainty  =  i  4.667  units. 

Length  of  the  earth's   quadrant  =  5131179*. 81  =  10000855'". 76 
Mean  uncertainty  =  =t  498'". 23 

-  7  '2  X.     4 

c  =  eccentricity  =  (  i  —  -----  J  =0.0816967 

log  =  8.9122052271 
E  =  ellipticity  =  i  e2  log  =  7.5233789824 

Length,  in  toises,  of  a  meridional  degree  whose  middle  lati 
tude  is  (?: 

D,H  =  57013^109  —  286*. 337  cos  2  <p  -f-  o'^n  cos  4  <p 

-f-  of.ooi  cos  6  <p 

Length  of  a  degree  of  the  parallel,  in  toises  : 

Dp  =  57 156*. 285  cos  <p  —  47t.825  cos  3  <p  -f-  ot.o6o  cos  5  <p 
or,  making  sin  »/'  =  e  sin  <f — 

log  D;)  =  4.7567009.0  -f-  log  cos  <p  —  log  cos  (," 


90  GEODESY. 


XXXVII.— Relative  Lengths  of  the  Yard  and  the  Metre. 

i.  -From  Clarke's  comparisons  referred  to  the  present  parlia 
mentary  standard,  (Comparison  of  Standards  of  Length  made  at 
the  Ordnance  Survey  Office  by  Captain  A.  R.  Clarke,  R.  E.,F.  R.  S., 
published  by  authority,  1866:) 

Values  Adopted  in  the  Measurement,  now  in  Progress,  of  an  Arc  of 
Parallel  Extending  from  Ireland  to  the  Rtier  Ural  in  Russia,  as 
"the  Exact  Relative  Lengths  of  Standards"  used  as  the  Units  of 
Measure  in  the  Triangulations  of  England,  France,  Belgium, 
Prussia,  and  Russia  : 


i  Expressed     in    Ex  din  |  Expressed  in 

Standards.  terms    of    the  i       .     ,  i    lines  of  the 


standard  yard. 


inches. 


toise. 


The  yard i .  oooooooo         36.  oooooo       405. 34622 

The  toise '     2.13151116          76.734402  i     864.00000 

The  metre 1.09362311          39.370432]     443.29600 


Expressed  in 
millimetres. 


914.39180 
1949.  03632 

IOOO. OOOOO 


]°g  39-37°432  =  i-59517°'lSl6 

2.   From  Rater's  comparisons  with  the  Shuckburg  scale,  (/%//. 
Trans,  for  1818:) 

i  metre  at  32°  F.  =  39.370790  inches  of  the  old  imperial  standard 
at  62°  F. 


3.  From  Hassler's  comparisons  of  the  Troughton  8  2  -inch 
scale,  with  the  iron  standard  committee  metre  of  the  American 
Philosophical  Society,  (Report  of  the  Secretary  of  the  Treasury  on 
the  Comparison  of  Weights  and  Measures,  Twenty  -Second  Congress, 
First  Session,  June  20,  1832  :) 

Value  adopted  by  the  United  States  Coast  Survey. 

i  metre  at    32°  F.  =  39.36850535  inches  of  the  Troughton  82- 

inch  scale  at  62°  F.  ' 


a  value  materially  smaller  than  the  preceding. 

There  is  a  doubt  whether  this  discordance  is  to  be  attributed  to 
inaccuracy  in  the  length  of  the  Troughton  scale  or  in  errors  in 
the  co-efficients  of  expansion  used  by  Mr.  Hassler. 


GEODESY.  91 


XXXVII.—  Relative  Lengths  of  the  Yard  and  the  Metre— Con'd. 

Logarithms  to  Reduce  Metres  to  Yards. 

Clarke.  Kater.  Coast  Survey. 

0.0388676809  0.0388716286  0.0388464579 

log  3  =  0.4771212547 

log   12  =  1.0791812460 

log  5280  =  3.7226339225 

Kater's  length  of  the  metre  in  English  inches  was  adopted  in 
the  preparation  of  the  first  edition  of  this  Collection  as  being  at 
the  time  most  generally  in  use.  It  has  been  retained  through 
out  the  present  volume,  although  the  results  of  Clarke's  compari 
sons  should  now  be  universally  adopted. 

The  committee  metre  of  the  American  Philosophical  Society  is 
the  unit  of  length  to  which  all  linear  measures  of  the  Coast  Sur 
vey  are  referred.  It  was  compared  August  24,  1867,  at  Paris, 
directly  with  the  standard  platinum  metre  of  the  Conservatoire 
des  arts  et  metiers,  and  was  found  (at  the  temperature  of  melting 
ice)  =  im.ooooo336  of  the  platinum  metre  of  the  archives. 

The  French  standard  metre  has  its  normal  length  at  zero  centi 
grade,  or  the  freezing-point.  It  was  intended  to  be  a  natural 
standard,  and  to -represent  the  ten-millionth  part  of  the  terrestrial 
arc  between  the  equator  and  the  pole,  which  was  assunred  to  be 
5130740  toises,  and  the  length  of  the  metre  443.29596  lines  of 
the  toise  du  Perou;  which  quantity  was  declared  by  law  in  1799 
to  be  the  length  of  the  legal  metre,  and  is  the  length  of  the  stand 
ard  platinum  metre  of  the  archives. 

The  toise  du  Perou,  made  in  1735,  is  a  bar  of  iron,  and  has  its 
standard  length  at  13°  Reaumur,  (6i°.25  Fahrenheit'.)  It  was 
used  by  La  Condamine  in  the  measurement  of  an  arc  of  the 
meridian  in  Peru  in  1744.  As  the  above  determination  of  the 
length  of  the  quarter  of  the  meridian  is  now  known  to  be  errone 
ous,  the  legal  metre  becomes,  in  fact,  but  a  legalized  part  of  the 
toise  du  Perou,  and  this  last  remains  the  primitive  standard.  It 
is  the  unit  of  length  in  which  the  greater  part  of  the  European 
geodetic  measurements  are  expressed. 

The  standard  Klafter  of  Vienna  has  its  normal  length  at  13° 
Reaumur,  and  is  =  840.76134  lines  of  the  toise  du  Perou. 

The  standard  Prussian  foot  is  a  standard  also  at  13°  Reaumur, 
and  was  declared  by  law  to  be  =  139.13  lines  of  the  toise  du 
Perou. 


92  GEODESY. 


XXXVIII.  —  Numerical  Values  of  BesseVs  Terrestrial  Elements  in 
English  Yards,  adopting  Rater's  Value-of  the  Metre,  viz  :  39.37079 
English  Inches  ;  log  1.5951  741  293. 

Log.  to  reduce  toises  to  yards  =  0.3286915586 
Log.  to  reduce  metres  to  yards  =  0.0388716286 

a  =  equatorial  radius  =  6974532^'.  339 
log  =  6.  8435150923 

l>  =  polar  radius  =695121  8^.059 
log  =  6.8420609125 

Length,  in  yards,  of  a  meridional  degree  whose  middle  lati 
tude  is  <p  : 

D,,(  =  1215255".  1  83  —  610^.336  cos  2  cr  4-  1^.302   cos  4  <r  ) 

-f  0^.002  cos  6  cr  j 

Length,  in  yards,  of  a  degree  of  the  parallel  : 
D^=  121830^.366  cos  (f  —  101^.941  cos  3  cr-fo>'.i2S  cos  5  <p 
or,  making  sin  y  =  e  sin  ^  — 

log  D,  =  5-°853925  +  log  cos  V  —  loS  cos  '/• 
or,  using  the  logarithms  of  the  numerical  co-efficients  — 
D,,,  =  121525^.183  —  (2.7855691)  cos  2  ^  -f  (0.1147)  cos  4  ^  > 

4-  (7.3287)  cos  6  <p  J 

Dy  =(5-°857556)  cos  V  —  (2-°°835)  cos  3  <F  +(9-Io69)  cos  5  V 
or  — 


D  ^-  cos  V 

COS    «/' 

Constant  Logarithms. 
r  =  0.00667435  ........................  log  =  7.8244104542 

£  ^  =  E  =  ellipticity  =  —  ---  =  7-5233789824 

299.66 

sin  i"  .............................  =  4-6855748668 

•J  Sin  i"  .....  :  ........................         =4.3845448711 

3  ^_sin  i7/  .............................         =2.6860751039 

•2 
(I  _  (*)  =  0.99332565  .............  =  9.9970916404 

0  (l  —  ^)  ............................       =  6.8406067325 

rt  sin  i/x  ............................  -  -         =  i-529o89959r 

a  sin  \"  ,  (arithmetical  complement)  .....  .  .         =  8.4709100409 


GEODESY. 


93 


XXXIX. — For  Computing  the  Geodetic  Latitudes,  Longitudes,  and 
Azimuths  of  Points  of  a  Triangulation. 

i.  For  distances  not  exceeding  one  hundred  miles : 
-  a  L  =  K  B  cos  Z  +  K2  C  sin2  Z  +  (8  L)2  D  -  K2  h  E  sin2  Z 
where — 

T 

B  = 


Rarci" 
tan  L 


2  N  R  arc  i/7 
y, f  e*  sin  L  cos  L  arc  \" 

(i  —  <?2  sin2  L)* 

_  i  +  3  Um'JjL    ' 
~6~N*~ 

h  =  K  B  cos  Z,  or  first  term  ; 

d  L  an  approximate  value  for  —  d  L,  computed  from  first  and 
second  term ; 

-V  K  sin  Z 
~cosL' 

; 

A  /  _  —  referred  to  second  point : 

N  arc  i" 

~~^  Z=         ^cosV^L 

log  F  for  latitude  25°  =  7.8324;  for  latitude  45°  =  7.8404. 

2.  For  distances  not  exceeding  twenty  miles : 

In  terms  of  the  sides  of  the  triangles : 


N  sin  i"  a  sin  i". 

L'  =  L  —  (i  +  e2  cos2  L)  u"  cos  Z  > 

-  (i  +  e1  cos2  L)  (H"  sin  Z)2  tan  L  X  J  sin  i"      f 


cos 


Z/=:i8o0  +Z.-'///--IJ-/Zsin  J  (L+  L') 
cos  L7 


94  GEODESY. 


XXXIX.  —  For  Computing  Gtodttic  Latitudes.  ,3-V.  —  Continued. 

In  terms  of  the  co-ordinates  of  rectangular  axes  referred  to  one 
of  the  points  of  the  triangulation,  the  latitude  and  longitude  of 
which  are  known  :  _r  being  the  ordinate  in  the  direction  of  the 
meridian,  and  x  the  ordinate  erendicular  to  it  : 


L  =L  =  ^-^-"(N^  y  *KL  =  R^T 

=         (N-  :•.:)•     -: 


K  =  distance  in  yards  between  two  station,  the  latitude  and 

:._---•"       -  .      :'  -    -   -  :  "      -    - 

distance  convened  to  seconds  of  arc  : 

L  =  latitude  of  ist  station: 

:  :='.:.-.'.      :">--'  -    : 

Z  =  azimuth  of  2d  station  at  ist,  counted  from  the  south  round 
by  the  west,  from  o°  to  360°:  the  algebraic  signs 
of  the  sine,  and  cosine  of  this  angle  must  be  carefully 

L7.  M',  Z/  the  same  things  at  2d  station,  or  quantities  required  : 


section  r*en>er.d:cuiar  to  the 


sn 


— 

(M'-M^sin  J; 

by  which  the  azimuth  at  one  end  of 
at  the  other,  is  called  the  cmrtrxfiK 


XL. — To  Compute  the  Length  and  Direction  of  a  Line  Joinmg  Two 
Points,  the  Latitudes  and  Longitudes  of  which  are  KiKncn,  or 
Measurement  of  a  Base  by  Astronomical  Observations. 

-.?  _  r  (L  —  L7)  cos2  ^  (L  + I/) 

~2~  2 

V=       . 


/=L'— 


=      —  .     —  i  sin  i  '  .v  -  tan  / 


-?      x 
tan  Z  = 


=  .r'  X  sin 


sin  Z      cos  Z 

_-.  =;.     X  sir.  : 
K  =  u"  X  sin  i 
in  which — 

L.  L' '.  M.  M7.  represent  the  latitudes  and  longitudes  of  the 
two  points : 

u".  the  distance  between  these  points  in  seconds  of  arc ; 
K.  the  distance  between  these  points  in  linear  units  : 

x",  the  number  of  seconds  in  the  arc  passing  through  the  point 
of  which  L7  is  the  latitude,  and  perpendicular  to  the 
meridian  of  the  point  of  which  L  is  the  latitude: 

\".  the  seconds  in  the  portion  of  this  meridian  between  L  and 
the  foot  of  this  perpendicular ; 

,\.  v,  the  same  quantities  in  linear  units  : 

Z.  the  azimuth  of  the  second  point,  L'.  from  the  first.  L ;  and 

N,  the  normal  at  the  middle  latitude. 


96 


GEODESY. 


XL. — To  Compute  the  Length  and  Direction  of  a  Line,  erV. — Con'd. 
Particular  attention  must  be  paid  to  the  sign  (L  —  L'),  for  upon 
this  depends  the  sign  of-  ,  and  also  to  that  of  (/ —  I'}  in  the 
value  of  y ,  so  as  to  know  whether  the   small  quantity — 

(— J  sin  i"#"2  tan /) 
is  to  be  added  to  or  substracted  from  (/  —  /'). 

The  azimuth  Z  is  counted  from  the  south,  round  by  the  west, 
from  o°  to  360°. 

The  azimuth  Z'   (if  required)   is  to  be  computed  from   Z,  as 
in  XXXIX,  (2.) 

This  can  be  presented  in  a  different  form,  thus  : 

as,  (M7  —  M)  cos  L7  =  u"  sin  Z 
and, 


r/j  y         •         rf/w       I        •"»  o   T    \ 

//          rj       \  -•-'       -"-*  j        y  **      out  t.j  \^\j^   \.j  liin  JL*  sin  i     \^\  — |—  t'~  cos*"  i^j 

Substituting,  in  this  last,  the  value  of  u"  sin  Z,  and  dividing 
one  by  the  other : 

(M'  -  M)  cos  L7  (i  +  el  cos2  L) 

~  (L-L')  -  j" (M/-M)2cos2  L7  tan  L  sin  i"  ( i  +  "^  cos8' L) 
Then,  knowing  Z — 

'       (M7  -  M)  cos  L7 
?/    =  — • — ^ — 

sin  Z 

and — 

K  =:  n"  N  sin  i7/ 
N  being  the  normal  for  the  mean  latitude. 

XLI. — To  Compute  the  Distance  between  Two  Points,  knowing  theit 
Latitudes  and  the  Azimuth  of  one  from  the  other. 


t3_e2  (L  -  L7)  cos2  j  (L  +  L7) 


e2  sin2  J  (L  +  L7)]i 


sin(^  —  //")  =  -=  —  j  sin  0  ; 

See  the  note  to  the  preceding  formulae.  The  algebraic  sign  of 
the  azimuth,  Z,  will  determine  the  sign  of  <p,  and  consequently 
whether  the  quantity  u"  is  to  be  added  to  or  subtracted  from  p. 


GEODESY. 


97 


XLII. —  To  Compute  the  Distance  between  Two  Points,  knowing 
the  Latitude  of  one,  the  Azimuth  from  this  to  the  other,  and  the 
Difference  of  their  Longitudes. 


tan  (  =  sin  L  tan  Z 


tan  L"  =  tan  L  sin  (y  ~ 

sin  <p 
-  L")  cos2  J  (L  +  L") 


L/  _  L//  _  .3  /  _  L  __  if  // j  /   i    ^ 

ui-i  ._  ;^cos-/  K  =  «"  N  sin  i/x 

sin  Z 

;;/  =  the  difference  of  longitude. 

The  azimuth,  Z,  is,  as  before,  counted  from  the  south  round  by 
the  west;  its  algebraic  sign  will  determine  the  sign  of  y,  and 
consequently  whether  it  is  to  be  increased  or  diminished  by  m. 


Elements  of  the  Figure  of  the  Earth,  Deduced  by  Captain  A.  R. 
Clarke,  Royal  Engineers,  in  Computing  the  Figures  of  the  Meri 
dians  and  of  the  Equator  for  Several  Measured  Arcs  of  Meri 
dian,  (Comparison  of  Standards  of  Length,  &c.,  London,  1866. ) 


Semi-axes. 


Major  semi-axis  =  «, 
34'  east)  

of  equator,  (longitude  15° 

Feet. 

Toiscs. 

Metres. 

Minor  semi-axis  =  <5, 
34'  east)  

of  equator,  (longitude  105° 

Polar  semi-axis  —  c.. 

6356068      I 

Length. 


a  —  c i 

c      = 285^97 


b—c 


a  —  b 


•313-38 


3269 -5 


The  length  of  the  meridian  quadrant  passing  through  Paris  is. 
and  the  minimum  quadrant,  in  longitude  105°  34',  is. 


,10001472.5  metres. 
.10000024.5  metres. 


For  a  Spheroid  of  Revolution  more  nearly  Corresponding  to  the  Same 
Geodetic  Measurements. 


Semi-axes. 


Length. 


Equatorial  semi-axis  —  a  

Feet. 
20926062 
20855121 

Toises. 

3272492.3 
3261398.4 

Metres  . 
6378206.4 
6356583-8 

Polar  semi-axis  —  b  .  . 

b       293.98                                   a  —  b            i 

llipticity. 

a  ~  294.  98                                          a      ~294.98~ 

98 


GEODESY. 


FORMS  FOR  RECORD 

Survey  of 


^6  3 

Position. 

Names   of 
stations. 

6 

Observed 
angles. 

14J 

^  .b  ~S 

_C    rC     £ 

"3    . 

'cLrt 

in 

Spherical 
excess. 

Final  plane 
angles. 

0              /                 // 

,, 

u 

n 

o          /           // 

Sought  . 

Cedar  Point.. 

18 

66  34  04.80 

—    O.36 

04.44 

1.58 

66  34  02.86 

XIII 

Right  .  . 

Buck  Hill  .  .  . 

18 

64  08  37.78 

-0.36 

37-43 

i.58 

64  08  35.84 

R 

• 

|| 

Left  .  .  . 

Fort  Flats  .  .  . 

18 

49  17  23.24 

-0.36 

'  22.88 

1.58 

j  47  17  21.30 

180    o    o.  oo 

Survey  of 


Names    of 
stations. 


Latitudes. 


L'  =  L-  u".(i  + 
sin  ix/  sin2  Z  «//2  ( 


os2  L)  cos  Z 
^-  cos2  L)  tan  L 


Fort  Flats . . 


Latitude  L  .......  =  45°  39'  i3"-! 


log  K  (yards). 


.=  4.7295212 
.=18.4701676 


log  u" =  3.1996888 

log  (i  +  ^3  cos2  L) =0.0014140 

log  cos  Z (  —  )  =  9.971 1240 


-4.38454 

2  log  sin  Z =  9.09522 

2  log  u" =  6.39936 


.  =  0.00141 


log  tan  L =  0.00991 


Cedar  Point  . 


log  1st  term  ..........  =  3.1722268    log  2d  term  .....  =  9.89034 

o;/.77 


ist  term 
2d  term 


(  +  )  =  +  i486".7i 
(  —  )  =  —       o".77 


2d  term 


L  ................  =  45°  39'  i3"-89    L+  Lf....=  9i    43  "3  -72 

Latitude  L'  .....  =  46°  03'  59"  -83    ~ 


GEODESY. 


99 


AND    COMPUTATION. 

Calculation  of  Triangles. 


bJ3 

P 

Logarithms 
of  their 
sines. 

Calculation  of 

the  sides. 

Sides    in 
yards. 

Designation. 

s 

R 
L 

9.9626198 
9.9541886 

9.8796760 

log  R  L 

—  4  737CK24. 

=  54695.61 
=  53644.00 

=  4518649 

$  Buck   Hill  —  Fort 
\      Flats. 

(  Fort  Flats—  Cedar 
\       Point. 

f  Buck  Hill—  Cedar 
\       Point. 

comp  log  sin  S  .  . 
log  sin  R  

—  o.  0373802 

=  9.9541886 

loo-  L  S 

=  4.7295212 

log  R  L  +          > 
comp  log  sin  S  ^ 
log  sin  L 

=  4-  7753326 
=  9.8796760 

loe  R  S 

=  4.6550086 

Geodetic  Determination  of  Positions. 


(Secondary/ 


Longitudes. 


Azimuths. 


Remarks. 


L  + 1/ ; 


Longitude  M  =^84°  42'  22".  19    Azimuth  Z =  159°  20'  i3". 


log  sin  Z (  +  )=9. 

log//' =  3. 


1 80° 


2.7473005 
log  cos  L' =  9.8412474 


log  6  M 


6  M 
M  .. 


.  =  2.9060529 


180°  -j-  Z =  339°  20'  13". 62 

20°  39'  46".38 


log  sin  L  +  L/ 


=  9-8559089 

..(  +  )  =  2. 9060529 


log  (5  Z 


.=12.7619618 


.=   o:  13'  25". 48 

.=  84°  42'  22".  19 


Longitude  M  =±84°  55'  47". 67 


r5Z 

180°  +  Z =  339°  20'  13".  62 


Azimuth  Z'  ..  ..  =  339°  10'  35". 5 7 


100                                                          GEOD'ESY. 

Normal  or  Radius  of   Curvature   of  the    Perpendicular    to    the 

Meridian. 

Ellipticity  =  ^o  ;  equatorial  radius  =  6974532  yards;  log  =  6.8435151. 

Log.  to  reduce  yards  to  metres  =  9.9611283714. 

N.=            a 

Latitude. 

(I—  ^  sin2  L)* 

log 
i+^cos'L) 

Difference 
for  10'. 

.logN 

Com.  difif. 
for  10'. 

lop-          * 

5  N  sin  i" 

20        0 

15 

3° 

45 

6.  8436847 
6888 
6929 
6971 

27-3 
27.6 
27.8 
28.1 

8.  4707404 
7363 
7322 
7280 

o.  0025521 
5439 
5356 
5274 

55 
.       56 
55 
56 

21         O 

7013 
7056 

28.4 

28    7 

7238" 
7196 

5106 

57 

30- 

7099 

zo-  / 

7153 

5021 

j?8 

45 

7142 

29-3 

7109 

4934 

58 

22         0 
15 

3° 

45 

7186 
7230 
7274 
7319 

29-5 
29-7 
30.0 
30.2 

7066 
7021 
6977 
6932 

4847 
4760 
4671 
4582 

58 
59 
59 
60 

23      o 
15 
30 
45 

7365 
7410 

7457 
7503 

30.5 
3°-7 
3I.O 
31.2 

6887 
6841 
6795 
6748 

4492 
4401 
43°9 
4217 

61 

62 
62 
62 

24      o 
15 
30 
45 

755° 
7597 
7645 
7693 

3*.  5 
31-? 

32.0 

32.2 

6701 
6654 
6607 

6559 

4124 
4°3° 
3935 
3840 

63 
63 

•s 

25      o 
15 

3° 

45 

774i 
7790 
7839 

,,000 

7000 

32.5 
32.7 
32-9 
33-2 

6510 
6462 
6413 
6363 

3744 
3648 
355° 
3452 

64 

65 

3 

26      o 
15 
30 
45 

7938 

7988 

8038 

33-4 
33.6 
33-8 
34-0 

6313 
6263 
6213 
6162 

3353 
3254 
3154 
3°53 

66 

67 
67 
68 

27      o 
15 
3° 
45 

8140 

8192 
8243 
8295 

34-3 
34.5 
34  7 
34-9 

6111 

6060 
6008 
5956 

2951 
2849 
2746 
2643 

68 
69 
69 
69 

28      o 

8348 

5904 

2539 

7° 

15 

8400 

5851 

2434 

70 

3° 

8453 

l\'\  :        5798 

2329 

45 

6.  8438507 

glj    8.4705745 

o.  0022223 

•" 

GEODESY. 


101 


Normal,  &c. — Continued. 


Latitude. 

N-     a 

log 
(i+ty2cos2L) 

Difference 
for  10'. 

(i  —  t>'2sin2  L)* 

logN 

Com.  diff. 
for  10'. 

log  L_ 

b  N  sin  i" 

0      / 

29   o 

15 

6.  8438560 
8614 

35.9 

8.  4705691 
5637 

o.  0022117 

2OIO 

71 

3° 

45 

8668 
8723 

36.  i 
36.3 
36.5 

5583 
5529 

1902 
1794 

72 
72 
72 

3°   ° 

8777 

oA  *T 

5474 

1686 

15 
3° 

45 

8832 
8888 
8943 

3°.  7 
36.9 
37.i 

37-2 

5419 
5364 
53°8 

1576 
1466 
1356 

73 
73 
73 
74 

31   o 
15 

8999 
'   9055 

37'6 

5252 

1245 
H34 

74 

3° 

45 

9111 
9168 

37.7 
37-9 

5084 

IO22 
0910 

75 
75 

32   o 
15 

9225 
9282 

38.1 
^8  -7 

5027 
4970 

0797 
0684 

75 

3° 

9339     Jj;~ 

4912 

0570 

n*r 

45 

4855 

0455 

/  / 

77 

33   o 
15 

3° 
45 

9454 
9512 

957i 
9629 

38.7 
38.9 
39-0  ' 
39-  i 

4797 
4737 
4681 
4622 

0340 
0225 
.  002OI09 
.0019993 

77 
77 
77 
77 

34   o 
15 
3° 
45 

9688 

9747 
9806 
9865 

39-2 
39-4 
39-5 
39-7 

4564 
45°5 
4446 

4387 

9877 
9760 
9043 
9525 

78 
78 
79 
79 

35   o 
15 

30 

45 

9924 
.  8439984 
.  8440044 
0104 

39-8 
40.  o 
40.  o 
40.  i 

4327 
4267 
4208 
4148 

9407 
9288 
9169 
9050 

80 
80 
80 
80 

36   o 
15 

3° 

0164 
0224 
0285 

40.3 
40.3 

4087 
4027 
3966 

8931 

8811 
8690 

80 
81 

45 

0346 

40!  6       3906 

8570 

81 

37   o 
15 
3° 

45 

0406 
0467 
0529 
0590 

40.7 
40.7 
40.9 
41.  o 

3845 
3784 
3g3 
3661 

8449 
8328 
8206 
8084 

81 
8r 
81 
81 

38   o 
15 

0651 
0713 

41.1 

AI  I  • 

3600 
3538 

7840 

82 

82 

3° 
45 

0775 
0837 

41  .  i 
41.2 
41-4 

3477 
3415 

7717 
7594 

82 
82 

0   0                            1 

39   ° 
15 

30 

45 

°8f     4I  4       3353 
0961              3291 
1023              3229 
6.8441085     41-5   8.4703166 

7471 
7348 
7224 
o.  0017101 

82 

11 

83 

102 


GEODESY. 


Normal,  &c. — Continued. 


N  = 

a 

Latitude. 

(  I  —  6'2  sin'2 

L)* 

-  log 

Difference 

logN 

Com.  diff. 
for  10'. 

**Nab<- 

(i+t'-cos2  L) 

for  10'. 

o     / 

40   o 

6.8441147 

A  T   *7 

8.  4703104 

o.  0016977 

15 

3° 

45 

I2IO 
1273 
1335 

41.  / 
41.8 
41.8 
41.9 

3°4i 
2979 
2916 

6853 
6728 

6604 

84 
83 
84 

41   o 
15 

3° 
45 

1398 
1461 

1524 

1587 

41.9 
41.9 
42.0 
42.1 

2853 
2791 
2728 
2665 

6479 
6354 

6229 
6104 

84 
84 
84 
84 

42   o 
15 
30 
45 

1650 

1713 

1776 

1839 

42.  I 
42.1 
42.2 
42.1 

2602 
2539 
2475 
2412 

5979 

5853 
5728 
5602 

84 
84 
84 
84 

43   ° 
15 

3° 
45 

1903 
1967 
2029 
2093 

42.2 
42.2 
42.3 
42.3 

2349 

2286 

2222 
2159 

5477 
535i 
5225 
5°99 

84 
84 
84 
84 

44   o 
15 

30 
45 

'2156 
2219 
2283 
2346 

42.3 
42.3 
42.3 
42.3 

2095 
2032 
1969 

4973 
4847 
4721 

4595 

84 
84 
84 
84  - 

45   ° 
15 

30 
45 

2410 
2473 

2537 
2600 

42.3 
42.3 
42.3 
42.3 

1842 
T778 

T5 

1651 

4469 

4343 
4217 
4091 

84 
84 
84 
84 

46   o 
15 
3° 
45 

2663 
2727 
2790 
2854 

42.3 
42.3 
42.3 
42.2 

1588 
1525 

1461 

1398 

3965 
3839 
37J3 
3587 

84 
84 
84 
84 

47   o 
15 

30 

45 

2917 
2980 
3043 

42.2 
42.  I 
42.1 

42.1 

1334 
1271 
1208 
1145 

3336 
3210 
3084 

84 
84 
84 

84 

48   o 
15 
3° 
45 

3170 

3233 
3296 

3359 

42.  I 

42.  o 

42.0 
41.9 

1082 

1018 

0955 
0892 

2959 

2833 
2708 

2583 

84 
84 
84 
84 

49   o 
15 
3° 
45 

3422 
3485 
3547 
3610 

41.9 

41.9 
41.8 
41.8 

0830 
0767 
0704 

0641 

2458 

2333 
2209 
2084 

84 
83 
84 
83 

50   o 

6.  8443673 

8.4700579 

o.  0011960 

GEODESY. 


I03 


Radius  of  Curvature  of  the  Meridian. 

Elliptieity  =  ^,-y  ;  equatorial  radius  =  6974532  yards. 


log  R 

Com.  diflf. 
for  10'. 

log  — 
33  Rsin  i" 

o      / 

20    0 
15 

6.8411155 

1278 

81.9 

82  7 

^2973 

3° 

45 

1402 

1527 

™*  / 

83.5 
84.3 

2849 
2724 

21    O         1654 

85  i 

2598 

15         1781 

86'.  o 

2470 

30         1910 

86^8 

2341 

45       2040 

2211 

22    0          2172 

88.4 

2080 

15          2304 
30          2438 

45      2573 

89.1 
90.0 
91.  o 

1947 
1679 

23   o 
15 

3° 

45 

2709 
2846 
2984 
3124 

91-5 
92.3 
93-° 
93-6 

1543 
H05 
1267 
1128 

24   o 
15 

3264 
3406 

94.6 

0987 
0845 

'3° 

3549 

ofi* 

07O2 

45       3693 

96.7 

°559 

25   o       3838 
15       3984 

97.4 

0414 
0268 

30       4I31 

yo.  i 

.  4730120 

45  ,     4279 

99.4 

.  4729972 

26   o 
15 

3° 

4428 
4578 
4730 

IOO.  I 

100.  9 
101,5 

9823 
9673 
9522 

45 

4882 

IO2.  I 

937° 

27   o 

5035 

102.8 

9216 

15 
30 

AC 

5l89 
5344 

103.4 

104.  I 

PC? 

43 

3D 

104.7 

75 

28   o 

15 

3° 

45 

5657 

5974 
6.8416134 

i°5-3 

106.  o 
106.5 
107.  i 

8594 
8436 
8277 
8.4728117 

io4 


GEODESY. 


Radius  of  Curvature  of  the  Meridian  —  Continued. 

Latitude. 

R^(7f^L)t 

log  R          Cfom-  di/f- 
for  10'. 

lo            ' 

Rsin  i  " 

29       o 
15 

6.  8416295 
6456 

107.  i 

8.4727956 
7795 

30 
45 

6619 
6782 

1  08.'  6 
109.  4 

7632 
7469 

30      o 

6946 

I  IO   O 

7305 

15 

30 

7111 

7277 

no.  5 

T  T  T       T 

7140 
6974 

45 

7444 

111.1 

in.  6 

6808 

31       o 

7611 

T  T  >•>       1 

6640 

15 

3° 

7779      ;;;•; 
7948 

6472 
6303 

45 

8118           JJ3<I 

32      o 
15 
30 
45 

8288 
8460 

88? 

114.  I 
114.  6 
115.1 
115.6 

5963 

5792 
5620 

5447 

33      ° 
15 
3° 

i   !ai 

5274 
5100 

4925 

45 

CKO2 

1  1  7.  3 

475° 

34      o 
15 
3° 

9678 
.  8419854 
.8420031 

117.7 
118.  i 

TTX     C 

4574 
4397 
4220 

45 

O2OQ                J 
.                  II9.I 

4042 

35      ° 

0387 

3864 

15 

3° 
45 

0566 
0746 
0926 

II9.7 

120.0 
120.4 

3685 
3325 

36      o 
15 
3° 

1107 

1288 
1469 

120.  7 

121.  I 

3H5 

2964 
2782 

45 

1651 

121.  7 

2600 

37      o 

'  1834 

2417 

15 

2017    !       IS  , 

2234 

3° 

22OO 

2051 

45 

2384 

122.7 

122.  9 

1867 

38      o 

15 

30 

45 

2569 
2753 
2939 
3124 

I23.I 

123.5 
123.7 

124.  o 

1683 
1498 

1313 
1127 

39      o 

3310 

0941 

15 

3496                     ^' 

0755 

3° 

3Oo7       :                                           O^Oo 

45 

6.  8423870           I24'  <-,       8.  4720^82 

1  2A«  O       1 

GEODESY. 


I05 


Radius  of  Curvature  of  the  Meridian — Continued. 


Latitude. 

R  = 

a(i-e*) 

a. 

(i—  ^2sin2  L)2 

log  R 

Com.  diff. 
for  10'. 

*nh» 

40   o 
15 
30 

45 

6.8424057 
4244 
4432 
4620 

125.0 
125.1 
125.3 
125.5 

8.4720194 
.  4720007 
.4719819 
9631 

41   o 

4808 

12$.  7 

9443 

3° 

4997 
5186 

:lo  !  ?* 

45 

5375 

126^2     8877 

42   o 
15 

5564 

5753 

126.2  !     f  7 

126  -     8498 

3° 

5943 

8709 

126.4                     Q°    ' 

45 

6132 

126.6 

43   ° 

6322 

126.6       7929 

15 

6512 

3° 

6702 

,'  '  ,     7549 

45 

6892 

^26  8       7359 

44   ° 
15 

3° 

7082 
7273 
7463 

5!  il 

45 

7653 

lll'.g       6598 

45   o 
15 
3° 
45 

7844 
8034 

8224 

8415 

127.  o 
126.  9 
126.9 
126.9 

6408 
6217 
6027 
5837 

46   o 
15 
30 

8605 

8795 
8985 

126.8 
126.8 
126  7 

5647 

5456 
5266 

45 

9175 

;:•  ' 
126.  7 

5076 

47   o 
15 
3° 
45 

9365 
9555 
9745 
.  8429934 

126.6 
126.6 
126.  4 
126.3 

4886 
4696 
4506 
43^7 

48   o 
15 

.8430124 
0313 

126.  2 
126.  I 

4127 
3938 

3° 
45 

0502 
0691 

126.0 

125.8 

3749 

49   ° 
15 

3° 

45 

0880 
1068 
1257 

1445 

125.7 
125.5 
125.3 
125.0 

3371 

3183 

2995 
2807 

50   o 

6.  8431632 

8.4712619 

io6 


GEODESY. 


XLIII. — Projection  of  Maps. 

POLYCONIC    PROJECTION. 

In  this  development  of  the  earth's  surface  each  parallel  of  lati 
tude  is  supposed  to  be  represented  on  a  plane  by  the  develop 
ment  of  a  cone  having  the  parallel  for  its  base  and  its  vertex  in 
the  point  where  a  tangent  to  the  parallel  intersects  the  earth's 
axis.  The  map  thus  becomes  the  development  of  the  surfaces  of 
several  successive  cones,  and  the  degrees  of  the  parallel  preserve 
their  true  length. 


Normal 


(i  — <?2  sin2  L)-v 

(i  —  e2} 
Radius  of  the  meridian ,         . .  R    =  N 3  -- 

a2 

Radius  of  the  parallel Ry,  =  N  cos  L 

Degree  of  the  meridian D  ,  =  -  *—  R/rt 

IOO 

=  3600  Rni  sin  i" 
Degree  of  the  parallel D/(  =  -  ,"  -  Rp 

=  3600  RyJ  sin  i" 
Radius  of  the  developed  parallel  or  side 

of  tangent  cone r  =  N  cot  L 

Designating  by  n  any  arc  of  the  parallel,  or  difference  of  longi 
tude,  to  be  developed,  and  by  0  the  corresponding  angle  sub 
tended  by  the  developed  parallel  at  the  vertex  of  the  cone;  then 
the  length  of  the  given  arc  will  be  : 

n  R^  =  //  N  cos  L 
and  also — 

Or  —  0  N  cot  L 
whence — 

Angle  of  the  developed  parallel,  0  =  n  sin  L 
and  as  the  developed  parallels  are  circular  arcs,  the  co-ordinates 
of  curvature  are  : 

JM)  difference  of  meridians,  =  x  =  r  sin  0 
dp,  difference  of  parallels,  =  y  =  r  ver  sin  0  =  x  tan  J  0 
For  surfaces  of  small  extent  the  arc  of  the  parallel  may  be  con 
sidered  coincident  with  its  chord ;  and   as  the  angle  between  a 


PROJECTION    OF    MAPS.  107 


XLIII. — Projection  of  Maps — Continued. 

tangent  and  a  chord  is  half  the  angle  at  the  center  subtended  by 
the  chord, 

dltl,  difference  of  meridians,  =  x  =  Dp  cos  J  0 
dp,  difference .  of  parallels,  =  y  =  T>p  sin  J  0 

The  values  of  dm  and  dp  and  of  Dw  and  D^  will  be  found  in  the 
following  tables. 

Example  of  their  Use. — Let  it  be  required  to  make  a  projection 
containing  40'  of  longitude  between  the  parallels  of  41°  30'  and 
42°  io7,  to  be  subdivided  to  5'. 

Assume  the  center  of  the  sheet  to  be  the  intersection  of  the 
middle  parallel  with  the  middle  meridian  of  the  proposed  map, 
which  point  call  A;  in  this  case  a  point  in  the  parallel  of  41°  50'. 

Through  A  draw  the  central  meridian  and  a  line  at  right 
angles  to  it. 

Beginning  at  A,  lay  of  above  and  below,  on  the  central  merid-. 
ian,  the  values  of  D,tt  from  41°  50'  to  41°  55';  41°  55'  to  42°; 
42°  to  42°  5',  etc.;  and  from  41°  50'  to  41°  45';  41°  45'  to 
41°  40',  etc. ;  these  values  to  be  taken  from  the  table  of  Meridional 
Arcs —  Values  of  Dm  in  Yards,  by  interpolation  from  the  values 
there  given  for  the  middle  latitudes  of  41°  and  42°. 

Through  each  of  the  points  . . . ,  A",  A1,  A,  Aj,  AH,  . . .  ,  thus 
found,  lay  off  perpendiculars  to  the  central  meridian. 

Now  turn  to  the  table  of  Co-ordinates,  OM  and  dp,  in  Yards,  and 
lay  off,  from  each  of  the  points  . . . ,  A",  A1,  A,  Aj,  Aii}  . . . ,  to  the 
right  and  left  of  the  central  meridian,  the  values  of  dm  for  succes 
sively  5',  10',  15',  and  20',  corresponding  (by  interpolation  from 
the  columns  of  41°  30'  and  42°)  to  each  parallel  of  latitude  re 
quired;  and,  from  the  points  thus  found,  the  corresponding  values 
of  dp  at  right  angles  to  the  lines  already  drawn. 

Lines  passing  through  the  extremities  of  dp  will  be  the  required 
meridians  and  parallels. 

The  projection  being  made,  any  point  whose  latitude  and  longi 
tude  are  known  will  be  projected  on  the  map  from  elements  taken 
from  the  tables  of  values  of  DMl  and  D^,  which  are  measured  from 
the  meridians  dj\&  parallels,  and  not  from  the  axes  of  co-ordinates 
used  in  making  the  projection. 


IOS                                                          GEODESY. 

Poly  conic  Projection  —  Co-ordinates,  din,  djt,  in  Yards. 

o 

Latitude  22°  o'. 

Latitude  22°  30'. 

Latitude  23°  o'. 

3 

'Sb 

c 

0 

4. 

*. 

4, 

,.,. 

4. 

4, 

I 

1882.0 

O.  I 

1875.3 

O.I 

1868.5             o.i 

2 

3763-9 

0.4 

3750.6 

0.4 

3737.0             0.4 

3 

5645-9 

0.9 

5625.9 

I.O 

5605.4 

I.O 

4 

7527.8 

1.6 

7501.2 

1.7 

7473-9 

1.7 

5 

9409.8 

2.6 

9376.4 

2.6 

9342.4 

2-7 

6 

11291.8 

3.7 

H25I.7 

3.7 

11210.9 

3-8 

7 

13173.7 

5.0 

I3I27.0 

5-1 

13079.4 

5-2 

8 

15055.7 

6.6 

I5OO2.3 

6.7 

14947.8 

6.8 

9 

16937.6 

8-3 

16877.6 

8-5 

16816.3 

8.6 

10 

18819.6 

10.3 

18752.9 

10.5 

18684.8 

10.6 

ii 

20701.6 

12.4 

20628.2 

12.6 

20553-3 

12.8 

12 

22583.5 

14.8 

22503.5 

15-0 

22421.8 

15.3 

13 

24465.5 

17-3 

24378.8 

17.6 

24290.2 

'17.9 

14 

26347.4               20.1 

26254.  I 

20.5 

26158.7 

20.8 

15 

28229.4               23.1 

28129.3 

23.5 

28027.2 

23.9 

16 

3OIII.4                26.2 

3OOO4.6 

26.7 

29895.7           27.2 

17 

31993.3                29.6 

31879.9 

30.2 

31764.2           30.7 

18 

33875.3 

33-2 

33755-2 

33.3 

33632.6           34.4 

19 

35757.2 

37.o 

35630.5 

37-7 

35501.1           38.3 

20 

37639.2 

41.0 

37505.8 

41.7 

37369-6 

42.5 

25 

47049.0 

64.1 

46675.8 

65.2 

46712.0 

66.4 

30 

56458.7 

92.3 

56258.7 

93-9 

56054.3 

95-6 

40 

75278.2 

164.1 

750H.5 

•    167.0 

74739-0 

169.9 

50 

94097.7 

256.3 

93764.2 

260.9 

93423.7 

265.5 

I      00 

II29I7.O 

369.1 

II25I6.9 

375-8 

112108.2 

382.3 

I       20 

150555.4 

656.2 

I5OO2I.9 

668.0 

149476.9 

679.6 

I       30 

169374.4 

830.5 

168774.2 

845.4 

168161.  i 

860.1 

I      40 

188193.3 

1025.4 

187526.3 

1043.8 

186845.1 

I  061.8 

2      OO 

225830.5 

1476.5 

225030.0 

1503-0 

224212.5 

1529.1 

2       30 

282284.7 

2307.1 

281284.0 

2348.5 

280261.9 

2389-2 

3     oo 

338736.6 

3322.2 

337535-6 

33SI.8 

336309.0 

3440.4 

3     30 

395186.0 

4521.9 

393784.5 

4603.0 

392353.i 

4682.8 

4    oo 

451632.0 

5906.2 

450029.9 

6012.1 

448393.7 

6116.2 

PROJECTION    OF    MAPS. 


109 


Poly  conic  Projection — Coordinates,  dm,  8p,  in  Yards. 


Longitude. 

Latitude  23°  30'. 

Latitude  24°  o'. 

Latitude  24°  30'. 

dm 

*, 

<*m 

8* 

dm 

&* 

i 

1861.5 

O.  I 

1854.4             o.i 

1847.2 

O.I 

2 

3723.1 

0.4 

3708.9                  0.4 

3694.4 

0.4 

3 

5584.6 

I.O 

5563-3             i.o 

554L6 

I.O 

4" 

7446.1 

1.8 

7417.7             1.8 

7388.8 

1.8 

5 

9307.6 

2.7 

9272.2             2.7 

9236.0 

2.8 

6 

11169.2 

3-9 

11126.6  !        3.9 

11083.2 

4.0 

7 

13030.7 

5-3 

12981.0 

5-4 

12930.4 

5.4 

S 

14892.2 

6.9 

14835-5 

7.o 

14777-6 

7-1 

9 

16753.7 

8.7 

16689.9 

8.9 

16624.8 

9.0 

10 

18615.3 

10.8 

18544-3 

II.  0 

18472.0 

11.  i 

ii 

20476.8 

13-1 

20398.8 

13.3 

20319.2 

13.5 

12 

22338.3 

15.6 

22253.2 

15-8 

22166.4 

16.0 

13 

24199.9 

18.2 

24107.6 

18.5 

24013.6 

18.8 

14 

26061.4 

21  .2 

25962.1 

21-5 

25860.8 

21.8 

15 

27922.9 

24-3 

27816.5 

24.7 

27708.0 

25-1 

16 

29784.4 

27.7 

29670.9 

28.1 

29555.2 

28.5 

17 

31646.0 

3L2 

31525.4 

31.7 

31402.4 

32.2 

18 

33507.5 

35-0 

33379-8 

35.5 

33249.6 

36.0 

19 

35369.0 

39-0 

35234.2 

39.6 

35096.8 

40.2 

20 

37230.5 

43-2 

37088.7 

43.9 

36944.0 

44-6 

25 

46538.1 

67.5 

46360.8 

68.6 

46179.9 

69.6 

30 

55845.8 

97.2 

55632.9 

98.7 

55415.9 

100.3 

40 

74460.9 

172.7 

74I77.I 

175.5 

73887.7 

178.3 

50 

93076.0 

269.9 

92721.3 

274-3 

92359.5 

278.5 

I      00 

111691.0 

388.7 

111265.3 

394-9 

110831.2 

401  .1 

I     20 

148920.6 

690.9 

148353-0 

702.  i 

I47774-I 

713-0 

I    30 

167535.2 

874.5 

166896.6 

888.6 

1*66245.4 

902.4 

I    40 

186149.7 

1079.6 

185440.1 

1097.0 

184716.5 

1114.  i 

2      00 

223377.9 

1554.6 

222526.4 

1579.7 

221658.0 

1604.3 

2      30 

279218.6 

2429.1 

278154-1 

2468.3 

277068.4 

2506.8 

3     oo 

335056.8 

3497-9 

333779-1 

3554-4 

332476.1 

3609.8 

3     30 

390892.0 

4761.1 

389401.1 

4837.9 

387880.6 

49!3-3 

4     oo 

446723.4 

6218.5 

445019.2 

6318.9 

44328i.i 

6417.4 

no 


GEODESY. 


Poly  conic  Projection — Co-ordinates,  dm,  df,  in  Yards. 


Longitude. 

Latitude  25°  o'. 

Latitude  25°  30'. 

Latitude  26°  o'. 

tJ« 

** 

6m 

6p 

6m 

SP 

0            / 

I 

1839.8 

O.  I 

IS32.3 

O.  I 

1824.7 

O.I 

2 

3679.6 

0.5 

3664.6 

0.5 

3649-3 

0.5 

•       3 

55I9-5 

I.O 

5496.9 

i  .0 

5474-0 

I.O 

4 

7359-3 

1.8 

7329.2 

1.8 

7298.6 

1.9 

5 

9199.1 

2.8 

9l6l.5 

2.9 

9I23-3 

2.9 

6 

11038.9 

4.1 

10993.8 

4.1 

10947.9 

4-2 

7 

12878.8 

5.5 

I2826.I 

5.6 

12772.6 

5.7 

8 

14718.6 

7.2 

14658.5 

7-3 

14597-2 

7-4 

9 

16558.4 

9-2 

16490.8 

9-3 

16421.9 

9-4 

10 

18398.2 

ii-3 

I8323.I 

ii.  5 

18246.5 

ii.  6 

ii 

20238.0 

13-7 

20155.4                13.9 

20071.2 

14.1 

12 

22077.9 

16.3 

21987.7                16.5 

21895.8 

16.8 

13 

23917.7 

19.1 

23820.0 

19.4 

23720.5 

19.7 

14 

25757-5 

22.2 

25652.3 

22.5 

25545-1 

22.  S 

15 

27597-3 

25-4 

27484.6 

25.8 

27369.8 

20.2 

16 

29437.1 

29.0 

29316.9 

29.4 

29194.4 

29.S 

17 

31277.0 

32.7 

3II49.2 

33-2 

31019.1 

33-C 

18 

33116.8 

36.6 

32981.5 

37-2 

32843-7 

37-  > 

19 

34956.6 

40.8 

34813.8 

41-4 

34668.4 

42.  c 

20 

36796.4 

45-2 

36646.1 

45-9 

36493.0 

46.; 

25 

45995-5 

70.7 

45807.6 

7L7 

45616.2 

72-: 

30 

55194.6 

101.8 

54969.1 

103.2 

54739-5 

104.- 

40 

73592.7 

180.9 

73292.0 

183.6 

72985-8 

186.1 

50 

91990.7 

282.7 

91614.9 

286.8 

91232.1 

290.5 

I       OO 

110388.6 

407.1 

109937.6 

413.0 

109478.3 

418.$ 

I       20 

147184-0 

723.8 

146582.7 

734-3 

145970.3 

744-  < 

I       30 

165581.5 

916.0 

164905.0 

929-3 

164216.0 

942.: 

I      40 

183978.8 

1130.9 

IS3227.I 

H47.3 

182461.5 

1163.. 

2      OO 

220772.7 

1628.5 

219870.6 

1652.1 

218951.9 

1675.1 

2      30 

275961.6 

2544-5 

274833.9 

2581.4 

273685.3 

2617.  ( 

3     °o 

33H47.8 

3664.1 

329794.3 

3717.3 

328415.8 

3769-: 

3     30 

386330.6 

4987.2 

38475L3 

5059.6 

383142.7  |     5130. 

4     oo 

441509.4 

6513-9' 

439704.0 

6608.5 

437865.3       6701.  ( 

PROJECTION    OF    MAPS. 


Ill 


Poly  conic  Projection  —  Co-ordinates  ^  dm,  dp,  in  Yards. 

0 

Latitude  26°  30'. 

Latitude  27°  o'. 

Latitude  27°  30  . 

«. 

*. 

<rro 

* 

0           / 

I 

1816.9 

O.I 

1808.9*            o.i 

1800.9 

O.  I 

2 

3633.7 

0.5 

3617.9  i           0.5 

3601.7 

0.5 

3 

5450.6 

i.i 

5426.8              i.i 

5402.6 

1.  1 

4 

7267.4 

1.9 

7235.7 

1.9 

7203.4 

1.9 

5 

9084.3 

2.9 

9044.6 

3.o 

9004.3 

3-0 

6 

10901.2 

4-2 

10853.6 

4-3 

10805.1 

4-4 

7 

12718.0 

5-8 

12662.5 

5-9 

12606.0 

5-9 

8 

14534.9 

7-5 

14471.4 

7-6 

14406.9 

7-7 

9 

16351-7 

9-5 

16280.3 

9-7 

16207.7 

9.8 

10 

18168.  6 

ii.  8 

18089.3 

11.9 

18008  6 

12.  r 

ii 

19985.5            14.3 

19898.2 

14.5 

19809.4 

14.6 

12 

21802.3            17-0 

21707.1            17.2 

21610.3 

17.4 

13 

23619.2            19.9 

23516.0               20.2 

23411.1 

20.4 

14 

25436.0           23.1 

25325.0               23.4 

25212.0 

23.7 

15 

27252.9           26.5 

2/133.9    j            26.9 

27012.8 

27.2 

16 

29069.8           30.2 

28942.8  :        30.6 

28813.7 

30.9 

17 

30886.6  ;         34.1 

3075L7           34-5 

30614.6 

34-9 

18 

32703.5           38.2 

32560.7           38.7 

32415-4 

39-2 

19 

34520.3           42.6 

34369.6           43-1 

34216.3 

43-6 

20 

36337.2 

47-2 

36178.5 

47-8 

36017.1 

48.4 

25 

45421.4 

73-7 

45223.1 

74-7 

45021.4 

75-6 

30 

54505.6 

106.1 

54267.7 

107.5 

54025.6 

108.8 

40 

72674.1 

188.6 

72356.8 

191.  i 

72034.0 

193-5 

50 

90842.4 

294.8 

90445  .  8 

298.6 

90042  .  4 

302.4 

I       OO 

109010.7 

424-5 

108534.8 

430.0 

108050.6 

435-4 

I       20 

145346.7 

754-6 

144712.1 

764.4 

144066.5 

774-0 

I       30 

163514.5 

955.1 

162800.5 

967.5 

162074.2 

979-6 

I      40 

181682.0  j     1179.  i 

180888.7 

1194.4 

180081.7 

1209.4 

2      OO 

218016.4 

1697.9 

217064.4 

1720.0 

216095.9 

1741-6 

2      30 

272515-9 

2652.9 

271325-7 

2687.5 

270114.9 

2721.2 

3      00 

327012.2 

3820.2 

325583-8 

3870.0 

324130.7 

3918.6 

3     30 

381505.0 

5199.8 

379838.2 

5267.5 

378142.5 

5333-6 

4     oo 

435993-2 

6791-  5 

434088.0       6880.0 

432149.7 

6966.3 

112 


GEODESY. 


Polyconic  Projection — Co-ordinates,  dm,  djt,  in  Yards. 


Longitude. 

Latitude  28°  o'. 

Latitude  28°  30'. 

Latitude  29°. 

<*«,  " 

* 

6m 

** 

dm 

6P 

i 

1792.7 

O.  I 

'1784-3 

O.  I 

1775-8 

O.  I 

2 

3585.3 

0.5 

3568.6 

0.5 

355L7 

0.5 

3 

5378.0  i     i.i 

5352.9 

i  .  i 

5327.5 

i.i 

4 

7I7O.6  I      2.O 

7I37-2 

2  .O 

7I03-3 

2  .O 

5 

8963.3  1     3-i 

8921.5 

3-1 

8879.1 

3-i 

6 

10755.9     4.4 

10705.8 

4-5 

10655.0 

4.5 

7 

12548.6  !    6.0. 

12490.2 

6,1 

12430.8 

6.1 

8 

14341.2      7-8 

14274-5 

7.9 

14206.6 

S.o 

9 

16133.9     9-9 

16058.8 

10.  0 

15982.5 

IO.  I 

10 

17926.5       12.2 

17843.1 

12.4 

17758.3 

12.5 

ii 

I97I9.2      14.8 

19627.4 

15-0 

10534-  i 

15.2 

12 

2I5II.8      17.6 

21411.7 

17.8 

21309.9 

iS.o 

13 

23304.5 

20.7 

23196.0 

20.9 

23085.8 

21.2 

14 

25097.1 

24.0 

24980.3 

24.3 

24861.6 

24-5 

15 

26889.8 

27.5 

26764.6 

27.9 

26637.4 

28.2 

16 

28682.4 

31.3 

28548.9 

31-7 

28413.2 

32.1 

17 

30475.1      35-4 

30333  •  2 

35-8 

30189.1 

36.2 

18 

32267.7      39.7 

32II7.5 

40.1 

31964.9 

40.6 

19 

34060.3'     44.2 

33901.8 

44-7 

33740.7 

45.2 

20 

35853.0      49-0 

35686.1 

49-5 

355i6.6 

50.1 

25 

44816.2  |     76.5. 

44607.6 

78.4 

44395-7 

78.3 

30 

53779-4 

no.  i 

53529.1    in.  4 

53274.8 

112.  7 

40 

71705.8 

195.8 

71372.1 

198.1 

71032.9 

200.3 

50 

89632.0 

306.0 

89214.9 

309.6 

88790.9 

313.0 

I   OO 

107558.2 

440.7 

107057.6 

445-8 

106548.8 

450.8 

I   20 

143410.0 

783-4 

142742.5    792.5 

142064.1 

801.4 

I   30 

I6J335.6  1   99r-5 

160584.6   1003.0 

159821.5 

1014.3 

I   40 

179260.9   1224.1 

178426.5 

1238.3 

177578.6 

1252  .  2 

2   00 

215110.9   1762.6 

214109.6   1783.2 

213092.0 

IS03.I 

2   30 

268883.6 

275-4-1 

267631.8   2786.2 

266359.6 

2817.4 

3  oo 

322652.8   3965-9 

321150.5   4012.1 

319623.7 

4057.1 

3  30 

376418.1   5398.1 

374665.0   5460.9 

372883.4 

5522.1 

4  oo 

430178.5  :  7050.6 

428174.6   7132  7 

426138.2 

7212.6 

PROJECTION    OF    MAPS. 


Poly co nic  Projection — Co-ordinates^  dm,  o  ,  in    Yards. 


Longitude. 

Latitude 

29°  30'. 
SP 

Latitude  30°  o'. 

Latitude  30°  30'. 

sm 

$m 

6* 

6m 

&* 

0     / 

I 

1767.2 

O.I 

1758.5 

O.I 

1749.6 

O.  I 

2 

3534-4 

0.5 

3516.9 

0-5 

3499.2 

0.5 

3 

5301.6 

I.I 

5275.4 

1.2 

5248.8 

I  .2 

4 

7068.9 

2  ,O 

7033.9 

2.O 

6998.3 

2.1 

5 

8836.1 

3-2 

8792.3 

3-2 

8747.9 

3-2 

6 

10603.3 

4.6 

10550.8 

4.6 

10497-5 

4.6 

7 

12370.5 

6.2 

12309.3 

6-3 

12247.  i 

6-3 

8 

I4I37.7 

S.i 

14067.8 

8.2 

13996.7 

8.2 

9 

15904.9 

10.3 

15826.2      10.4 

15746.3 

10.5 

10 

17672.2 

12.7 

17584.7     12.8 

17495-9 

12.9 

TI 

19439-4 

15-3 

I9343-I  ;    15-5 

19245-4 

15.6 

12 

21206.6 

lS.2 

21101.6     18.4 

20995.0 

18.6 

13 

22973.8 

21.4 

22860.1  |    21.6 

22744.6 

21.8 

14 

24741.0 

24.8 

24618.5    25.1 

23494-2 

25-3 

15 

26508.2 

28.5 

26377.0  i   28.8 

25243-8 

29.1 

16 

28275.4 

32.4 

28135.5  j   32.7 

26993-3 

33-1 

17 

30042.6 

36.6 

29893-9     37-0 

28742.9 

37-4 

iS 

31809.9 

41.0 

31652.4 

41.4 

30492.5 

41.8 

19 

33577-1 

45-7 

33410.9 

46.2 

32242.1 

46.6 

20 

35344-3 

5  1-  -6 

35169.3 

51-2 

3399!-7 

51-7 

25 

44180.3 

79.1 

43961.6 

,79-9 

43239-6 

80.7 

30 

53016.4 

II3-9 

52753.9 

115.1 

52487-4 

116.3 

40 

70688.4 

202.5 

70338.4 

204.6 

69983.  i 

206.6 

50 

88360.2 

316.4 

87922.8 

3I9-7 

87478.7 

322.9 

I   00 

106032.0 

455-6 

105507.1 

460.4 

104974.1 

464-9 

I  20 

I4I375.0 

810.0 

140675-1 

818.4 

139964.4 

826.6 

I  30 

159046.2 

1025.2 

158258.7 

1035-8 

I57459-2 

1046.1 

I  40 

176717.1 

1265.7 

175842.2 

1278.8 

174953.8 

1291.5 

2   00 

212058.1 

1822.6 

211008.  i 

1841.5 

209942.  i 

1859.8 

2   30 

265067.  i 

2847-8 

263754.5 

2877.3 

262421.8 

2905  .  9 

3  oo 

318072.5 

4100.8 

316497.1 

4143.3 

314897.6 

4184.5 

3  30 

371073.4 

5581.7 

369235-2 

5639-5 

367368.9 

5695-6 

4  oo 

424069.3 

7290.3 

421968.0 

7365.9 

419834.7 

7439-1 

GEODESY. 


Polyconic 

Projection  —  Co-ordinates,  3m,  dp,  in  Yards. 

a> 

"O 

d 

Latitude  31°  o'. 

Latitude  31°  30'. 

Latitude  32°  o'. 

'So 

6m                 6P 

c 

0 

6m 

6P 

6m 

** 

i' 

1740.6 

O.  I 

I73I-4 

O.  I 

1722.  I 

O.  I 

2 

348I.I 

0.5 

3462.8 

0.5 

'3444-3 

0.5 

3 

5221.7 

I  .2 

5194.3 

1.2 

5166.4 

1.2 

4 

6962.3 

2.1 

6925.7 

2.1 

6888.6 

2.  I 

5 

8702.9 

3  •  3 

8657.1 

3  •  3 

8610.7 

3  •  3 

6 

10443.4 

4-7 

10388.5 

4-7 

10332.8 

4.8 

7 

12184.0 

6.4 

12119.9 

6.4 

12055  .0 

6.5       ' 

8 

13924.6 

8-3 

13851-4 

8.4 

I3777-I 

8-5 

9 

15665.1 

10.6 

15582.8 

10.7 

15499-3 

10.8 

10 

17405.7 

13.0 

17314.2 

13-2 

17221.4 

13-3 

ii 

19146.3 

15-8 

19045.6 

15-9 

18943-6 

16.1 

12 

20886.8 

18.8 

20777.1 

18.9 

20665.7 

19.1 

13 

22627.4 

22.  O 

22508.5 

22.2 

22387.8 

22.4 

14 

24368.0 

25-6 

24239.9 

25.8 

24110.0 

26.0 

15 

26108.5 

29-3 

2597L3 

29.6 

25832.1 

29.9    . 

16 

27849.1 

33-4 

27702.7 

33-7 

27554-3 

34-0 

1 
17 

29589.7 

37-7 

29434.2 

38.0 

29276.4 

38.4 

18 

31330.3 

42.2 

31165.6 

42.6 

30998.5 

43-0 

19 

33070.8 

47-i 

32897.0 

47-5 

32720.7 

47-9 

20 

34811.4 

52.2 

34628.4 

52.6 

34442.8 

53-1 

25 

43514.2 

8i.f 

43286.0 

82.3 

43053.5 

83.0 

30 

52217.0 

II7-3 

51942.5          118.4 

51664.1 

II9-5 

40 

69622.5 

208.6 

69256.6 

210.  5 

68885.4 

212.4 

50 

87027.9 

326.0 

86570.5 

328.9 

86106.5 

331-8 

I       OO 

104433.2 

469-4 

103884.3 

473-7 

103327.4 

477-8 

I       2O 

139243.1 

834-5 

138511.2 

842.1 

137768.8 

849-5 

I       30 

156647.8 

1056.1 

155824.4 

1065.8 

154989.1 

1075.1 

I       40 

174052.2 

1303-8 

173137.2 

1315-8 

172209.1 

I327-3 

2       00 

208860.0 

I877-5 

207762.0 

1894.7 

206648.2 

1911.3 

2       30 

261069.  i 

i     2933.7 

259696.5 

2960.5 

258304.1 

2986.5 

3     oo 

313274.2 

I     4224.5 

311626.9 

4263.1 

309955.S 

4300.5 

3     30 

365474.6 

5750.0 

363552.4 

5802.6 

361602  ..5 

5853.5 

4     oo 

417669.4 

i     7510.2 

415472.3 

7578.9 

4I3243-4 

PROJFXTION    OF    MAPS. 


Poly  conic  Projection  —  Co-ordinates,  dm,  djt,  in'  Yards. 

0> 
T3 

3 

1 

0 

hJ 

Latitude  32°  30'. 

Latitude  33°  o'. 

Latitude 

33°  30'. 

<J* 

3* 

dm 

*, 

6m 

<** 

I 

1712.7 

O.  I 

1703.2 

O.I 

1693-5 

O.I 

2 

34250 

0.5 

3406.4 

0.5 

3387-0 

0.5 

3 

5138.2 

I  .2 

5  i  09  .  6 

1.2 

5080.6 

1.2 

4 

6850.9 

2.1 

6812.8 

2.2 

6774.1 

2.2 

5 

8563-7 

3-3 

8515-9 

3-4 

8467.6 

3-4 

6 

7 

10276.4 
11989.  i 

4.8 
6.6 

10219.  i 
11922.3 

4-9 
6.6 

10161.  i 
18154.6 

4-9 
6.7 

8 

13701.8 

8.6 

13625.5 

8.6 

I3548.I 

8-7 

9 

15414.6            10.  S 

15328.7 

10.9 

15241-7 

II.  0 

10 

17127.3            13.4 

17031.9 

13-5 

16935.2 

13-6 

ii 

18840.0           16.2 

18735-1 

16.3 

18628.7 

16.4 

12 

20552.8  ,      19.3 

20438.3 

19.4 

20322.2 

19.6 

13 

22265.5    '            22.6 

22141.4 

22.  S 

22015.7 

23.0 

14 
15 

23978.2                26.2 
25690.9               30.1 

•  23844.6 

25547.8 

26.4 
30-4 

23709-2 
25402.7 

26.6 
30.6 

16 

27403.7            34-3 

27251.0 

34-5 

27096.3 

34-9 

17 

29116.4            38.7 

28954.2 

39-0 

28789.8 

39-3 

18 

30829.1            43.4 

30657,4 

43-7 

30483-3 

44-0 

19 

32541.9  i         48.5 

32360.6 

49.0 

32176.8 

49.1 

20 

34254.6  :      53-5 

34063.8 

54.0 

33870.3 

54-4 

25 

42818.2        83.6 

42579.6 

«4-3 

42337-9 

85.0 

30 

51381.8       120.5 

51095.5 

121.4 

50805.4 

122.3 

40 

68508.9  ;     214.1 

68127.2 

215.9 

67740.4 

217-5 

50 

85635.9 

334-6 

85158.8 

337-3 

84675-2 

339-9 

I      OO 

102762.7 

481.8 

102190.2 

485.7 

101609.9 

489.4 

I     20 

137015.8 

856.6 

136252.4 

863.5 

135478.6 

870.1 

I    30 

154142.0 

1084.1 

153283.1 

1092.8 

152412.6 

IIOI.2 

I    40 

171267.9 

1338.4 

170313.6 

1349-2 

169346.3 

1359-5 

2       OO 

205518.7 

1927.4 

20*373.5 

1942.8 

203212.7 

1957-7 

2       30 

256892.0     3011.5 

255460.4 

3035-6 

254009.2 

•3058.8 

3     oo 

308261.1   ;     4336.6 

306542.9 

437L3 

304801.4 

4404  •  7 

3     30 

359625.1        5902.  6 

357620.3 

5949-9 

355588.2 

5995-3 

4     oo 

410983.2        7709-5 

408691.6 

7771.2 

406368.8 

7830.6 

116                                                          GEODESY. 

Poly  conic   Projection  —  Co-ordinates,  din,  dp,  in  Yards. 

<J 

T3 

a 

Latitude  34°  o'. 

Latitude  34°  30'. 

Latitude  35^  o'. 

'So 

j 

6m 

> 

Sm 

*, 

6m 

** 

0            1 

I 

1683.7 

O.  I 

1673.8 

O.  I 

1663.7 

,     o.i 

2 

3367.4 

0.5 

3347-6 

0.6 

3327.5 

0.6 

3 

505LI 

1.2 

5021.4 

I  .  2 

4991.2 

1.2 

4 

6734.9 

2.2 

6695.1 

2.2 

6654.9 

2.2 

5 

8418.6 

3-4 

8368.9 

3-4 

8318.7 

3-5 

6 

IOI02.3 

4-9 

10042.7 

5-0 

9982.4 

5-o 

7 

II786.0 

6-7 

11716.5 

6.8 

11646.  i 

6.8 

8 

13469.7 

8.8 

13390.3 

S.8 

13309.8 

8.9 

9 

I5I53.4 

II.  I 

15064.1 

II.  2 

14973-6 

II  .2    . 

10 

16837.2                13.7 

16737.9 

-     13-8 

16637.3 

13-9 

ii 

18520.9           16.6 

18411.6 

I6.7 

18301  .0 

16.8 

12 

20204.6           19.7 

20085.4 

19.9 

19964.8 

20.  o 

13 

21888.3           23.1 

21759-2 

23-3 

21628.5 

23-5 

14 

23572.0           26.8 

23433-0 

27.0 

23292.2 

27.2 

15 

25255-7           30.8 

25106.8 

31.0 

24955-9 

31.2 

16 

26939-4           35-1 

26780.6 

35-3 

26619.7 

35.5 

17 

28623.1 

39-6 

28454-4 

39-8 

28283.4 

40.1 

18 

30306.9           44.4 

30128.1 

44-7 

29947.1 

45.o 

IQ 

31990-6           49-4 

31801.9  ! 

49-8 

31610.9 

50.1 

20 

33674.3           54-3 

33475-7 

55-2 

33274.6 

'  55-5 

25 

42092.8           85.6 

41844-6 

86.2 

4I593.2 

86.7 

30 

50511.4          123.2 

50213.5 

125.1 

49911-8 

124.9 

40 

673^8.3 

219.1 

66951.1 

220.  6 

66548.9 

222.1 

50 

84185.1 

342.3 

83688.7 

344-7 

83185.8 

.     347-0 

I       00 

IOIO2I  .8 

493-0 

100426.0 

496.4 

99822  .  6 

499-7 

I     20 

134694.5 

876.4 

I33900.I     ! 

882.5 

133095.5 

888.3 

I    30 

151530.5  1    1109.2 

150636.8 

1116.9 

I4973L5 

1124.2 

I   40 

168366.1  !   1369.4 

I67373.I 

1378.9 

166367.3 

1387-9 

2      OO 

202036.4 

I97L9 

200844.7 

1985.6 

199637.7 

1998.6 

2      30 

252538.7 

3081.1 

251049.0 

3102.5 

249540.  i 

3122.8 

3     oo 

303036.6 

4436.8 

301248.7 

4467.5 

299437.8 

4496.9 

3     30 

353529.0      6039.0 

351442.8 

6080.8 

349329.8 

6120.8 

4    oo 

404015.1      7887.7 

401630.5     i 

7942.3 

399215.4 

7994-5 

PROJECTION    OF    MAPS. 


117 


Poly  conic  Projection — Co-ordinates,  dm,  o2>,  in   Yards, 


3 
4 
5 
6 

7 
8 

9 
10 

!  I 
12 
13 

'4 
15 

16 

17 
18 

J9 

2; s 

25 
5'» 
40 
50 

oo 

20 
30 
40 
2   OO 

2  30 

3  oo 
3  30 


4  oo 


Latitude  35°  30'. 

Latitude  36°  o'. 

Latitude  36°  30'. 

4. 

4, 

1 

"<•: 

4, 

1653-5  i          o.i 

1643.2  i          o.i 

1632.8 

O.I 

3307.1             0.6 

3286.5             0.6 

3265.6 

0.6 

4960.6             1.3 

4929-7 

i-3 

4898.4 

i-3 

6614.2    !               2.2 

6572.9                  2.2 

6531-2 

2-3 

8267.7         3.5 

8216.2                  3.5 

8164.0 

3-5 

9921.3          5.0 

9859-4  '           5-i 

9796.8 

5-r 

11574.8             6.8 

II502.7                  6.9 

11429.6 

6.9 

13228.4             8.9 

I3I45.9                  9.0 

13062.4 

9.0 

14881.9           11.3 

14789.1   :         ii.  4 

14695.2 

ii.  4 

16535.5            14.0 

16432.4            14.0 

16328.0 

14.1 

18189.0           16.9 

18075.6  i         17.0 

17960.8 

17.0 

19842.5 

20.  i 

19718.8               20.2 

19593-6 

20.3 

21496.1 

23-6 

2I362.O               23.7 

21226.4            23.9 

23149.6 

27.4 

23005.3    1            27.5 

22859.2            27.7 

24803.2 

31-4 

24648.5               31,6 

24492.0 

31.  S 

26456.7 

35-7 

26291.8               36.0 

26124.8            36.2 

28110.3 

40.4 

27935-0               40.6 

27757.6            40.8 

29763.8 

45-2 

29578.2               45.5 

29390.4            45.8 

3I4I7.3 

50.4 

3T22I.5 

50.7 

31023.2            51.0 

33070.9 

55-9 

32864.7 

56.2 

32656.0  i          56.5 

41338.6 

87.2 

4IOSO.S 

87.8 

40819.9            88.3 

i 

49606  .  2 

125.7 

49296.9 

126.4 

48983.9 

127.1 

66141.5 

223.4 

65729.1 

224.8 

653H.7 

226.0 

82676.6 

349-1 

82l6l.I 

351-2 

81639.3 

353-1 

992II.5 

502.7 

98592.9 

505.7 

97966.7 

508.5 

132280.7 

893-8 

I3M55.9 

899.1 

130621.0 

904.1 

148814.9 

1131.2 

147886.9 

II37-9 

146947.7 

1144.2 

165348.8 

1396.6 

164317.7 

1404.8 

163274.0 

1412.6 

198415.4 

2011.  I 

197178.0 

2022.9 

195925.6 

2034.1 

2480I2.I 

3M2.3 

246465.3 

3160.8 

244899.6 

3178.3 

297604.0 

4524-9 

295747.6 

4551-6 

293868.6 

4576.8 

347190.2    |      6158.9 

345024.1 

6195.2 

342831.7 

6229.5 

396769.7    '       8044.3 

394293.8          8091.6 

391787.8 

8136.5 

Il8                                                         GEODESY. 

Foly  conic  Projection  —  Co-ordinates,  <JM,  djt,  in  Yards. 

c.' 

'U 

Latitude  37°  o'. 

Latitude  37°  30'. 

Latitude  38°  o'. 

c 

0 

«.. 

*, 

«. 

* 

«. 

* 

0            / 

' 

I 

1622.2 

O.  I 

1611.6 

O.  I 

1600.7  |          o.i 

2 

3244.5 

0.6 

3223.1 

0.6 

3201.5             0.6 

3 

4866.7 

i-3 

4834.7 

i-3 

4802.2 

i-3 

4 

6489.0 

2-3 

6446.2 

2-3 

6403.0 

2-3 

5 

8111.2 

3-5 

8057.8 

3.6 

8003.7 

3.6 

6 

9733-4 

5.i 

9669.3 

5-i 

9604.5 

5-2 

7 

H355.7 

7.0 

11280.9 

7-o 

11205.2 

7-0 

8 

12977-9 

9.1  . 

12892.4 

9.1 

12806.0 

9-2 

9 

14600.2 

ii.  5 

14504-0 

ii.  6 

14406.7 

ii.  6 

10 

16222.4 

14.2 

16115.6 

I4o 

16007.5 

14-3 

ii 

17844.6 

17.2 

I7727.I 

17.3 

17608.2 

17-3 

12 

19466.9 

20.4 

19338.6 

20.5 

19209.0 

20.6 

13 

21089.1 

24.0 

20950.2 

24-1 

20809.7 

24.2 

14 

22711.4 

27.8 

22561.8 

28.0 

22410.5 

28.1 

15 

24333-6 

31-9 

24173.3 

32.1 

24011.2 

32.3 

16 

25955-8 

36.4 

25784-9 

36.5 

25612.0           36.7 

17 

27578.1 

41.0 

27396.4 

41.2 

27212.7 

41.4 

18 

29200.3 

46.0 

29008  .  o 

46.2 

28813.4 

46.4 

19 

30822.6 

51-3 

30619.5 

51.5 

30414.2 

51.7 

20 

32444.8 

56.8 

32231.  i 

57-1 

32015.0 

57.3 

25 

40555.9 

88.7 

40288.8 

89.1 

40018.6 

89.6 

30 

48667.1 

127.8 

48346.5 

128.4 

48022.3 

129.0 

40 

64889.2 

227.2 

64461.9 

228.3 

64029.6 

229.3 

50 

81111.3 

355-0 

80577.1 

356.7 

80036.7 

358.3 

I       OO 

97333-1 

SIT-  2 

96692.0 

513.7 

96043  -  6 

516.0 

I       20 

129776.1 

908.8 

128921.3 

913.2. 

128056.7 

917.4 

I       30 

145997-2 

1150.2 

145035.5 

II55-8 

144062.8 

1161.0 

I       40 

162217.9 

1419.9 

161149.4 

1426.9 

160068.5 

1433-4 

2      OO 

194658.2 

2044.7 

193375-9 

2054-7 

192078.9 

2064.  i 

2      30 

243315.2 

3194.9 

241712.2 

3210.5 

240090.8 

3225.1 

3     oo 

291967.  i 

4600.7 

290043.4 

4623.1 

288097.5 

4644.1 

3     30 

340613.1 

6262.0 

338368.4 

6292.6 

336098.0 

6321.2 

4    oo 

389251-9 

8178.9 

386086.3 

8218.8 

384091.2 

8256.3 

PROJECTION    OF    MAPS.                                             119 

Polyconic  Projection  —  Co-ordinates,  o,,(,  o,.,  in  Yards. 

4 

Latitude  38°  30'. 

Latitude  39°  o'. 

Latitude  39°  30', 

3 

c 

0 

Oro 

SP 

rfm 

6p 

<Jm 

dp 

0            / 

I 

1589.8    i 

O.I 

1578.8 

O.  I 

1567.6 

O.I 

2 

3179.6             0.6 

3157.5 

0.6 

3135.2 

0.6 

3 

4769-5              r-3 

4736.3 

i.3 

4702.8 

1-3 

4 

6359.3             2,3 

63I5.I 

2-3 

6270.4 

2-3 

5 

7949.1             3-6 

7893.8 

3.6 

7838.0 

3-6 

6 

9538.9  :        5.2 

9472.6 

5-2 

9405.6 

5.2 

7 

11128.7 

7.1 

H05I.4 

7-i 

10973.2 

7-i 

8 

12718.6 

9.2 

12630.  I 

.    9.2 

12540.8 

9-3 

9. 

14308.4         11.7 

14208.9 

ii.  7 

14108.4 

ii.  7 

10 

15898.2        14.4 

I57S7.7 

14-5 

15676.0 

14-5 

ii 

17488.0         17.4 

17366.4 

17-5 

17243.6 

17-5 

12 

19077.8           20.7 

18945.2 

20.8 

iSSn.i 

20.9 

13 

20667.6           24.3 

20524.0 

24-4 

20378.7 

24-5 

14 

22257.4 

28.2 

22IO2.7 

28.3 

21946.3 

28.4 

15 

23847-3 

32.4 

23681.5 

32.5 

235TJ-  9 

32.6 

16 

25437.1           36.8 

25260.3 

37.0 

25081.5 

37-1 

17 

27026.9           41.6 

26839.0 

41.8 

26649.1 

41.9 

18 

28616.7 

46.6 

28417.8 

46.8 

28216.7 

47-0 

19 

30206.5 

52.0 

29996.6 

52.2 

29784.3 

52.3 

20 

31796.3 

57-6 

31575.3 

57-8 

3I35L9 

58.0 

25 

39745-4 

90.0 

39469.1 

90.3 

39189.8 

90.6 

30 

47694.4 

129.5 

47362.9 

130.1 

47027.7 

130.5 

40 

63592.4 

230.3 

63150.3 

231.2 

62703.4 

232.0 

50 

79490.2 

359-9 

78937.6 

361.3 

78379.0 

362.6 

I      OO 

95387.8 

518.2 

94724.7 

520.2 

94054.4 

522.1 

I       20 

127182.3 

921.2 

126298.  I 

924.8 

125404-3 

928.2 

I       30 

143079.0 

1165.9 

142084.4 

1170.5 

141078.8 

1174.7 

I      40 

158975.5 

1439-4 

157870.3 

I445-I 

156753.0 

1450.2 

2      OO 

190767.1 

2072.8 

189440.8 

2080.9 

188100.1 

2088.3 

2      30 

238451.0       3238.7 

236793.0 

325L4 

235116.9 

3263.0 

3     oo 

286129.6       4663.8 

284139.8 

4682.0 

282128.3 

4698.8 

3     30 

333801.8       6347.9 

331480.2 

6372.7 

329133-2 

6395.6 

4     oo 

381466.7  1     8291.2 

3788I3.I 

8323.6 

376130.5 

8353.4 

120 

GEODESY. 

Polyconic 

Projection  —  Co-ordinates,  3IH)  <5JJ}  in  Yards. 

o 
T3 

Latitude  40°  o'. 

Latitude  40°  30'. 

Latitude  41°  o'. 

'So 

3 

3,* 

** 

dm 

*, 

6m 

&P 

I 

1556.3       ; 

O.  I 

1544-9 

O.I 

1533-4 

O.I 

2 

3II2.6 

0.6 

3089.8 

0.6 

3066.7 

0.6 

3 

4668  .  9 

1-3 

4634.7 

i-3 

4600.  I 

1-3 

4 

6225.2 

2-3 

6179.6 

2  >  3 

6133.5 

2-3 

5 

7731.5 

3-6 

7724.5 

3-6 

7666.8 

3-7 

c 

9337-8 

5-2 

9269.4 

5-3 

9200.2 

5-3 

7 

10894.1 

7-1 

10814.3 

7-2 

10733.6 

7-2 

8 

12450.4 

9-3  • 

12359-2 

9-3 

12266.9 

9-4 

9 

14006.7 

u.  8 

13904.0            1  1.  8 

13800.3 

11.  g 

10 

15563-0 

14.6 

15448.9 

14.6 

15333.7 

14.6 

ii 

17119.3 

17.6 

16993.8 

17-7 

16867.0 

17-7 

12 

18675.6 

21  .O 

18538.7 

21  .O 

18400.4 

a  1.  1 

13 

20231.9 

24.6 

20083.6 

24-7 

19933-7 

24-7 

14 

21788.2- 

28.5 

21628.5 

28.6 

21467.1 

28.7 

15 

23344-5 

32.7 

23173.4 

32.8 

23000.4 

32.9 

16 

24900.8 

37-2 

24718.3 

37-4 

24533.8 

37-5 

17 

26457.1 

42.0 

26263.2 

42.2 

26067.2 

42-3 

18 

28013.4 

47.1 

27808.1 

47-3 

27600.5 

47-4 

19 

29569.8 

52.5 

29352.9 

52.7 

29133-9 

52.8 

20 

31126.1 

58.2 

30897.8 

58.4 

30667.3 

58.5 

25 

38907-5 

90.9 

38622.2 

91.2 

38334.0 

91.4 

30 

46689.0 

130.9 

46346.7          I3I.3 

46000  .  8 

I3L7 

40 

62251.8 

232.8 

6I795.3          233.5 

61334.2 

234.1 

50 

77814.4 

363-7 

77243.9  :       364-8 

76667.4 

365.8 

I       OO 

933/6-9 

523-8 

92692.2          525-3 

92000.4 

526.7 

i     20 

124500.9 

931.2 

123588.0  (       933-9 

122665.7 

936.4 

i     30 

140062.5 

1178.5 

I39035.5        1182.0 

137997.8 

1185.1 

i     40 

155623.7 

1455-0 

154482.6 

1459.3 

I53329-6 

1463.1 

2       OO 

186744.9 

2095.2 

185375.4 

2101.4 

183991-8 

2106.9 

2       30 

233422.9 

3273.7 

231710.9 

3283.4 

229998.3 

3292.0 

3     oo 

280095.3 

4714.1 

278040.9  |     4728.1 

275965-1 

4740.6 

3     30 

326761.1 

6416.5 

324364.1        6435.4 

321942.2 

6452.4 

4     oo 

3734I9-3 

8380.7       370679.4  '     8405.5        367911.3 

8427-7 

PROJECTION    OF    MAPS. 


121 


Poly  conic  Projection  —  Co-ordinates,  dm,  Stt1  in  Yards. 

0) 
"O 

Latitude  41°  30'. 

Latitude  42°  o'. 

Latitude  42°  30'. 

Si) 

o 

J 

d»*                  $v 

6m 

6P 

dm 

* 

I 

1521.7              o.i 

I5IO.O 

O.  I 

1498.1 

O.  I 

2 

3043.4              0.6 

3019.9 

0.6 

2996.2 

0.6 

3 

4565-2              1-3 

4529.9 

i-3 

4494.2 

i-3 

4 

6086.9              2-3 

6039.8 

2.4 

5992.3 

2.4 

5 

7608.6              3.7 

7549-8 

3-7 

7490.4 

3-7 

6 

9*30.3              5-3 

9059.7 

5-3 

8988.5 

5-3 

7 

10652.0              7.2 

10569.7 

7-2 

10486.5 

7-2 

8 

12173.8              9.4 

12079.6 

9-4 

11984.6 

9-4 

9 

13695.5            11.9 

13589.6 

n.  9 

13482.7 

n.  9 

10 

15217.2            14.7 

15099.6 

14.7 

14980.8 

14.7 

ii 

16738.9            17.7 

i  6609  .  5 

17.8 

16478.8 

17.8 

I  2 

IS260.6               21.  I 

18119.5 

21  .2 

17976.9 

21.2 

13 

19782.3               24.8 

19629.4 

24.8 

19475-0 

24-9 

14 

21304.0               28.7 

21139.4 

28.8 

20973.1 

28.8 

15 

22825.8               33.0 

22649.3 

33-1 

22471.1 

33-1 

16 

24347-5        •   37-5 

24159-3 

37-6 

23969-2            37-6 

17 

25869.2           42.4 

25669.2 

42.5 

25467-3 

42.5 

18 

27390.9           47.5 

27179.2 

47-6 

26965.4 

47-7 

19 

28912.6           52.9 

28689.1 

53-0 

28463.4 

53-i 

20 

30434-3           58.6 

30199.1 

58.8 

29961.5 

58.9 

25 

38042.9           91.6 

3774S.8 

91.8 

37451-3 

92.0 

30 

45651.4          132.0 

45293.5 

132.3 

44942.2 

132.5 

40 

60868.3         234.6 

60397.8 

235-1 

59922.7 

235-5 

50 

76085.1          366.6 

75496-9 

367-4 

74903.0 

368.0 

I       00 

91301.6         527.9 

90595-9 

529.0 

89883.2 

529-9 

I       20 

121733.9         933.6 

120792.9 

940.5 

119842.6 

942.1 

I     30 

136949.6    *  1187.9 

135890.9 

1190.3 

134821.8 

1192.3 

i     40 

152164.9        1466.5 

150988.5 

1469.5 

149800.6 

1472.0 

2      00 

182594.1          2III.8 

181182.5 

2116.1 

179756.9 

2119.7 

2      30 

228234.1         ;               3299.7 

226469.4 

3306.4 

224687.4 

3312.0 

3     oo 

273868.3                       4751-6 

271750.5 

4761.2 

269612.0 

4769-3 

•  3     30 

319495.6                       6467.5 

317024.7 

6480.5 

3I4529.5 

6491  .6 

4    oo 

365115.0  !    8447.3 

362290.8 

8464.4 

359438.9 

8478.8 

122 

GEODESY. 

Polyconic 

Projection  —  Co-ordinates,  dm,  djt,  in  Yards. 

3 

'So 

c 

•   2 

Latitude  43°  o'. 

Latitude  43°  30'. 

Latitude 

44°  o'. 

*, 

6m 

* 

*, 

j 

I 

1486.1 

O.  I 

1474.0 

O.  I 

1461.8 

O.I 

2 

2972.2 

0.6 

2948.0 

0.6 

2923-5 

0.6 

3 

4458.3 

1-3 

4421.9 

i.3 

4385.3 

1.3 

4 

5944.3 

2.4 

5895.9 

2.4 

5847.0 

2.4 

5 

7430.4 

3-7 

7369-9 

3-7 

7308.8 

3-7 

6 

8916.5 

5-3 

8843.9 

5-3 

8770.5 

5-3 

7 

10402.6 

7-2 

10317.8 

7-2 

10232.3 

7-2 

8 

11888.7 

9-4 

11791.8 

9-5 

11694.1 

9-5 

9 

13374-8 

11.9 

13265.8 

12.0 

I3I55.8 

12.0 

10 

14860.8 

14.7 

14739-8 

I4.8 

14617.6 

I4.8 

ii 

16346.9 

I7.S 

16213.7 

I7.S 

16079.3 

17.9 

12 

17833.0 

2T.2 

17687.7 

21.2 

I754LI 

21.3 

13 

19319.1 

24.9 

19161  .7 

24-9 

19002.8 

25.O 

14 

20805.2 

28.9 

20635.7 

28.9 

20464.6 

28.9 

15 

22291.2 

33-2 

22109.6 

33-2 

21926.3 

33-2 

16 

23777-3 

37.7 

23583.6 

37.8 

23388.1 

37-8. 

17 

25263.4 

42.6 

25057-6 

42.6 

24849.9 

42.7 

18 

26749-5 

47-8 

26531.6 

47-8 

26311.6 

47-9 

19 

28235.6 

53-2 

28005.5 

53-3 

27773-4 

53-3 

20 

29721.7 

59-o 

29479.5 

59-o 

29235.1 

59-1 

25 

37102.0 

92.1 

36849-3 

92.2 

36543.8 

92.3 

30 

44582.4 

132.7 

44219.2 

132.8 

43852.6 

132.9 

40 
50 

59443-0 
74303.4 

235-8 
368.5 

58958.7 
73698.0 

236.  i 
368.9 

58469.9 
73087.0 

236.3 
369-2 

I      OO 

89163.6 

530.7 

88437.1 

531-2 

87703.9 

53L7 

I     20 
I    30 

118883.1 
133742.4 

943-4 
1194.0 

H79M-5 

132652.7 

944-4 
H95.3 

116936.9 
I3I552.9 

945-2 
1196.3 

I     40 

148601.2 

1474.1 

I47390.5 

1475-7 

146168.4 

1476.9 

2      OO 

178317.6 

2122.7 

176864.7 

2125.0 

175398.2 

2126.7 

2      30 

222888.2 

3316.7 

221071.9 

3320.3 

219238.7 

3323-0 

3     oo 

267452.8 

4776.0 

265273.1 

4781.3 

263073.1 

4785.1 

3     30 

312010.3 

6500.7 

309467.1 

6507.9 

306900.3 

6513-0 

4     oo 

356559.5 

8490.7 

353652.8 

8500.1 

350719-° 

8506-8 

PROJECTION    OF    MAPS. 

'I23 

Poly  conic  Projection  —  Co  ordinates,  dm,  dlt,  in  Yards. 

o> 

TD 

3 

1 

O 
J 

Latitude  44°  30'. 

Latitude  45°  o'. 

Latitude 

45°  30'. 

6m 

*, 

dm 

*, 

<U 

6* 

o        / 

I 

1449.4 

O.  I 

t437.0 

O.I 

1424.4 

O.  I 

2 

2898.9 

0.6 

2874.0 

0.6 

2848.9 

0.6 

3 

4348.3 

i-3 

4311.0 

,    1.3 

4273.3 

1-3 

4 

5797-7 

2.4 

5747-9 

2  .  4 

5697.7 

2.4 

5 
6 

7247.1 
8696,6 

3-7 
5-3 

7184.9 
8621  .9 

3-7 
5-3 

7122.2 
8546.6 

3-7 
5-3 

7 

10146.0 

7-2 

10058.9 

7-2 

9971.0 

7-2 

8 

11595.4             9.5 

11495.9 

9-5 

II395.4 

9-5 

9 

13044.8 

12.  O 

12932.9 

12.0 

12819.9 

12.0 

10 

14494-3 

I4.8 

14369.8 

14-8 

14244.3 

I4.8 

ii 

15943-7 

17.9 

15806.8 

17-9 

15668.7 

17-9 

12 

I7393-I 

21.3 

17243.8 

21.3 

17093.1 

21-3 

13' 

18842.5 

25.O 

18680.8 

25.0 

18517.6 

25.0 

14 

20292.0 

29.0 

20117.7 

29.0 

19942.0 

20.  O 

15 

21741.4 

33-2 

21554-7 

33-3 

21366.4 

33-2 

16 

23190.8 

37-8 

22991.7 

37-8 

22790.8 

37-8 

17 

24640  .  2 

42.7 

24428.7 

42-7 

24215.3 

42.7 

18 

26089.7 

47-9 

25865-7 

47-9 

25639.7 

47-9 

19 

27539-1            53-3 

27302.7 

53-4 

27064.1 

53-3 

20 

28988.5            59.1 

28739.6 

59-1 

28488.6 

59-1 

25 

36235.6           92.3 

35924.5 

92.4 

35610.6 

92-3 

30 

43482.6 

133-0 

43109.3 

i33-o 

42732.7 

i33.o 

40 

57976.6         236.4 

57478.9 

236.5 

56976.8 

236.4 

50 

I      00 
I       20 
I       30 

72470.4         369-4 
86964.0         531.9 
115950.3    !        945.7 
130442.9   [      1196.8 

71848.3 
86217.4 

H4954-9 
129323.0 

369-5 
532-0 
945-8 
1197.1 

71220.6 
85464-2 
II3950.6 
128193.2 

369-4 
532.0 

945-7 
1196.9 

I       40 

144935.2 

1477.6 

143690.8 

1477.9 

142435.5 

1477-7 

2      OO 

i739l8-3 

2127.7 

172425.0 

2128.1 

I709I8.5 

2127.8 

2      30 

217388.7 

3324.6 

215522.0 

3325.2 

213638.8 

3324.8 

3     oo 

260853.0       4787.4 

258612.0 

4788.3 

256352.9 

4787.7 

3     30 

304310.0       6516.2 

301696.3 

6517-3 

299059.6 

6516.5 

4     oo 

347758.4 

8510.9 

344771-2 

8512.5 

34r757.6 

8511.4 

124                                                             GEODESY. 

Poly  conic  Projection  —  Co-ordinates,  dm,  dp,  in  Yards. 

6 

Latitude  46°  o'. 

Latitude  46°  30'. 

Latitude  47°  o'. 

fc/i 
c 

,3 

4. 

*, 

4,       |       * 

4. 

v 

I 

I4II.8 

O.  I 

1399.0 

O.I 

I386.I 

O.  I 

2 

2823.5 

0.6 

2798.0 

0.6 

2772.2 

0.6 

3 

4235.3 

i-3 

4197.0 

1-3 

4153.4 

1.3 

4 

5647.1 

2-4 

5596.0 

2.4 

5544-5 

2.4 

5 

7058.8 

3-7 

6995.0 

3-7 

6930.6 

3-7 

6 

8470.6 

5-3 

8394.0 

5-3 

8316.7 

5-3 

7 

9882.4 

7-2 

9793-0 

7-2 

9702.8 

7-  2 

8 

II294.  I 

9-5 

11192.0 

9-4 

IloSg.O 

9-4 

9 

12705.9 

12.  O 

12591.0 

12.  O 

I2475-I 

II.  9 

10 

I4H7.7 

I4.8 

13990.0 

,    I4.8 

I386I.2 

14.7 

ii 

15529.4 

17.9 

15389-0 

17.9 

15247.3 

17.8 

12 

16941.2 

21.3 

16788.0 

21.3 

16633.4 

21.2 

13 

18353.0 

25.0 

18186.9 

24.9 

18019.5 

•24.9 

14 

19764.7 

28.9 

19585-9 

28.9 

19405.7 

28.9 

15 

2II76.5 

33-2 

20984.9 

33-2 

2O79I.8 

33-2 

16 

22588.3 

37.8 

22383.9 

37-8 

22177.9 

37-7 

17 

24000.0 

42.7 

23782.9. 

42.7 

23564.0 

42.6 

18 

254II.8 

47-9 

25181.9 

47-8 

24950.1 

47.8 

J9 

26823.6 

53-3 

26580.9 

53-3 

26336.2 

53-2 

20 

28235.3 

59-1 

27979.9 

59-0 

27722.4 

59.0 

25 

35294.1 

92-3 

24974-8 

92.2 

34652.9 

92.2 

30 

42352.9 

132.9 

41969.8 

132.8 

41583.4 

132.7 

40 

56470.3 

236.3 

55959-5 

236.2 

55444-3 

235.9 

50 

70587.5 

369-3 

69949-0 

369.0 

69305.1 

368.6 

I       OO 

84704.5 

531.7 

83938.2 

531-3 

83165.6 

530.8 

I       20 

II2937-6 

945-3 

111915.9 

944-6 

110885.7 

943-6 

I       30 

127053.6 

1196.4 

125904.2 

II95-5 

124747.2 

II94-3 

I      40 

I4II69.2 

1477-0 

139892.1 

1476.0 

138604.3 

1474.4 

2      OO 

169399.0 

2126.9 

167866.4 

2125.4 

166321.0 

2123.2 

2      30 

2H739-3 

3323.3 

209823.5 

3320.9 

207891.7 

3317.5 

3     oo 

254073.4 

4785.6 

251774.4 

4782.1 

249456.0 

4777-2 

3     30 

296400.0 

6513-8 

293717.6 

6509.0 

291012.8 

6502.7 

4     oo     338717.8        8507.8        335652.1 

8501.5 

332560.6 

8492.7 

PROJECTION    OF    MAPS. 


I25 


Poly  conic  Projection — Co-ordinates,  dm,  8  ,  in  Yards. 


03 

-a 

_a 

Latitude  47°  30'. 

Latitude  48°  o'. 

» 
Latitude  48'  30'. 

| 
2 

*• 

&P 

8m 

£ 

<*« 

«*, 

0           / 

I 

I373-I 

O.I 

1360.0            o.i 

1346.9 

O.  I 

2 

2746.3 

0.6 

2720.1             0.6 

2693.7 

0.6 

3 

4119-4 

L3 

4080.1              1.3 

4040.6 

1-3 

4 

5492.5 

2.4 

5440.2             2.4 

5387.4 

2.4 

5 

6865.7 

3-7 

6800.2             3.7 

6734.3 

3-7 

6 

8238.8 

5-3 

8160.3             5-3 

8081.  i 

5-3 

7 

9612.0 

7.2 

9520.3             7.2 

9428.0 

7-2 

8 

10985.1 

9.4 

10880.4             9.4 

10774.8 

9-4 

9 

12353    2 

ii.  9 

12240.4            ii.  g 

12121.7 

ii.  9 

TO 

I373L4 

.4-7 

13600.5  !         14.7 

13468.5 

14.7 

II 

15104.5 

I7.8 

14960.5            17.8 

14815.4 

17.8 

12 

16477.6 

21.2 

16320.5    I            21.2 

16162.2 

21.  I 

13 

17850.8 

24.9 

17680.6    |            24.8 

17509.1 

24.8 

14 

19223.9 

28.9 

19040.6               28.S 

18855.9 

28.8 

15 

20597.0 

33-1 

20400.7               33.1 

20202.8 

33-0 

16 

2I970.I 

37-7 

21760.7 

37-6 

21549.6 

37-6 

17 

23343-3 

42.6 

23120.7 

42.5 

22896.5 

42.4 

18 

24716.4 

'     47-7 

24480.8 

47-6 

24243.3 

47-5 

T9 

26089.5 

53-1 

25840.8 

53-1 

25590.2 

53-0 

20 

27462.7 

58.9 

27200.9 

58.8 

26937.0 

58.7 

25 

34328.3 

92.0 

34001.0 

91.9 

33671.2 

91.7 

30 

41193.9 

132.5 

40801.2 

132.3 

•40405.4 

132.0 

40 

54925.0 

235-6 

54401.4 

235-2 

53873.6 

234.7 

50 

68655.8 

368.1 

68COI.4    |         367.5 

6734L7 

366.8 

I       00 

82386.5 

530.1 

8l6oi.  I 

529.2 

80809.  5 

528.2 

I     20 

109846.9 

942.4 

108799.7 

940.8 

107744.2 

939-° 

I    30 

123576.6 

1192.7 

I22398.5 

1190.7 

I2I2II.O 

1188.4 

I    40 

137305.8 

1472.4 

135996.8 

1470.0 

134677.3 

1467.1 

2       OO 

164762.8 

2120.3 

163191  .9 

2116.  8 

161608.  6 

2112.7 

2       30 

205943-9 

33I3.0 

205980.3 

3307.5 

2O2OOI.O 

330I.I 

3     oo 

247118.6 

4770.7 

244762.2 

4762.9 

242387.0 

4753-5 

3     30 

288285.6 

6493.5 

285536.3    i      6482.8 

282765.2 

6470.  i 

4     oo 

329443.7    | 

8481.3 

326301.5          8467.3 

323134.3 

8450-7 

126 


GEODESY. 


Poly  conic  Projection — Co-ordinates,  dm,  8p;  in  Yards. 


Td 

3 

Latitude  49°  o'. 

Latitude 

49°  30'. 

Latitude  .50°  o'. 

C 
0 

*. 

* 

dm 

dp 

6m 

* 

I 

1333.6 

O.  I 

1320.2 

O.I 

1306.7             o.i 

2 

2667.1 

0.6 

2640.3 

0.6 

2213.3             °.6 

3 

4000.7 

i-3 

2960.5 

i-3 

3920.0             1.3 

4 

5334-2 

2-3 

5280.6 

2-3 

5226.6 

2-3 

5 

6667.8 

3-7 

6600.8 

3-7 

6533.3             3.6 

6 

8001.3 

5-3 

7920.9 

5-3 

7839-9             5-2 

7 

9334-9 

7.2 

9241.1 

7-2 

9146.6 

7-1 

8 

10668.4 

9-4 

10561.2 

9-3 

10453.2             9.3 

9 

I2OO2.O                  II.Q 

11881.4 

ii.  8 

11759.9            n.  8 

10 

13335-5         .     T4-6 

13201.6 

14.6 

13066.5            14.6 

ii 

14669.1 

17.7 

I452I.7 

17.7 

14373.2            17-6 

12 

16002.7 

21.  I 

15841.8 

21.  0 

15679-8 

21.0 

13 

17336.2              24.7 

17162.0 

24.7 

16986.5 

24.6 

14 

18669.7  \           28.7 

18482.1 

28.6 

18293.1 

28.5 

T5 

20003.3              32.9 

19802.3 

32-8 

19599.8 

32.8 

16 

21336.8 

37-5 

21122.4 

37-4 

20906  .  4 

37-  3 

17 

22670.4              42.3 

22442.6 

42.2 

22213.1 

42.1 

18 

24004.0              47.4 

23762.8 

47-3 

235I9.7 

47.2 

19 

25337-5 

52.8 

25082.9 

52.7 

24826.4 

52.6 

20 

26671.1              58.6 

26403.  i 

58.4 

26133.0 

58.2 

25 

33338.8              91-5 

33003.8 

91.2 

32666.2           91.  c 

30 

40006.5  -         131.7 

39604.5 

I3L4 

39199.4          131.0 

40 

53341-7            234.2 

52805.7 

233.6 

52265.7         232.  (] 

50 

66676.8            365.9 

66006  .  8 

365-0 

6533T-7          364-c 

I       00 

80011.6            527.0 

79207.6 

525-6 

78397.5          524-1 

I       20 

106680.4            936.8 

105608.3 

934-4 

104528.2         931.7 

I       30 

120074.2          1185.7 

118808.2 

1182.6 

II7593-0        H79-- 

I       40 

133347.5   ,       1463-8 

132007.5 

1460.0 

130657.4 

I455-? 

2      CO 

160012.8          2107.9 

158404.8 

2102.5 

156784.7 

2096.4 

2       3O 

200006.3          3293-6 

197996.2 

3285.1 

195970.8 

3275-f 

3     oo 

239993-2          4742.8 

237581.0 

4730.5 

235150.6 

4716.  c 

3     30 

279972.3          6455.4 

277158.0 

6438.8 

274322.4 

6420.2 

4     oo 

319942.3          8431.6 

316725.8 

8409.9 

313485.0 

8335-  I 

PROJECTION    OF    MAPS. 


127 


Arcs  of  Para  He!—  Values  of  D,,  in  Yards. 


L.  20°  30'. 

L.  21°  0'. 

L.  21°  30'. 

L.  22°  O'.  ;  L.  22°  30'. 

L.  23°  o7. 

/   // 

7 

221.8 

221.  I 

220.  3 

219.6    218.  8 

218.0 

8 

253.5 

252.6 

251.8 

250.  9     250.  o 

249.  i 

9 

285.2 

284.  2 

283.3 

282.3     281.3 

280.3 

10 

316.9 

315.8 

S'4-7 

3I3.7     312.5 

311.  4 

2O 

633.7 

631.6 

629.5 

627.3     625.1 

622.8 

3° 
40 

950.6 
1267.4 

947-4 
1263.  2 

944-2 
1259.0 

941.0     937.6 
1254.6    1250.2 

934.2 
1245.  7 

5° 

1584.3 

I579-  1 

1573-7 

1568.3    1562.7 

60 

1901.  1 

1894.9 

1888.  5 

1881.9    1875.3 

1868.5 

7  oo 

13307-  8 

13264.  i 

13219.4 

I3I73-7   13127.0 

T3079-4 

8  oo 
9  oo 

10  OO 
20  00 

30  oo 
40  oo 

15208.  9 
17110.0 
19011.  i 
38022.  i 
57033.2 
76044.  3 

15159.0 

1  7053-  8 
18948.  7 

37897.  4 
56846.  i 
75794.8 

15107.9 
16996.  4 
18884.  9 

37769.  7 
56654.6. 

75539-  5 

i5°55-  7   15002.3 
16937.6   16877.6 
18819.  6  ;  18752.  9 
37639.2  ;  37505.8 
56458.  8   36258.  8 
75278.4  '  75011.7 

M947-  9 
16816.3 
18684.  8 
37369.6 
56054.  4 
74739-  3 

50  oo 
60  oo 

95055.4 
114066.4 

94743-  4 
113692.  i 

94424-  3 
113309.2 

94098.  o   93764.  6 
112917.  7  \  112517.6 

93424.  i 
112108.  9 

Meridional  Arcs — Values  of  I)w  ///  Yards. 


i 

L.  21  °0'. 

L.  22°  O;. 

L.  23°  o'. 

8 

235-4 

269.0 

235.4 
269.  I 

235-5 
269.  i 

9 

302.  7 

302.  7 

302.8 

10 

336.3 

336.4 

336.4 

20 

672.6 

672.7 

672.8 

30 

1008.  9 

1009.  i 

• 

1009.  2 

40 

1345.  2 

I345.4 

r345-6 

lo 

- 

1  681.6 

2017.9 

1  681.8 

2018.  i 

1682.0 
2018.4 

14125.0 

14126.  7' 

14128.5 

8  oo 
9  oo 

16142.  9 
18160.8 

16144.  8 
18162.  9 

16146.8 
18165.2 

10  oo 
20  oo 
30  oo 
40  oo 

20178.6 
40357.2 

60535.  9 
80714.5 

20181.  o 
40362.  i 
60543.  i 
80724.  i 

20183.  5 
40367.  i 
60550.  6 
80734.  i 

50  oo 
60  oo 

100893.  i 
121071.  7 

100905.  2 
I2I086.  2 

100917.  6 

I2IIOI.  2 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal  point. 


128                                                            GEODESY. 

Arcs  of  Parallel  —  Values  of  D^,  in 

Yards. 

L.  23°  30'.     L.  24°  o'. 

L.  24°  30'. 

L.  25°  o'. 

L.  25°  30'. 

L.  26°  o'. 

217.2 

216.4 

215.4 

214.  6 

213.8 

212.  9 

g 

248.2 

247.3 

246.3 

245-3 

244-3 

243-3 

9 

279.2 

278.2 

•    277.  i 

276.  o 

274.8 

273.7 

10 

309.  i 

307.9 

306.6 

305.4 

304.1 

20 

620.5 

618.  i 

613-3 

610.8 

608.  2                  . 

30 
40 

£ 

930.8 
1241.  o 

927.6 
1236.  3 

1545-4 
1854.4 

923.6 
1231-5 

1539-3 

1847.  2 

919.9 
1226.5 
1533-2 

1839.8 

916.2 

1221.  5 
1526.9 
1832.3 

9I2.3 
I2I6.4 
1520.5 

1824.  7 

7  oo 
8  oo 
9  oo 
10  oo 

20   00 

30  oo 
40  oo 
50  oo 
60  oo 

13030.7 

14892.  2 

16753.  8 

18615.3 

37230.  6 
55845.8 
74461.  i 
93076.  4 
111691.  7 

12981.  o 

14835-  5 
16689.9 

18544-3 
37088.  7 

55633.  o 
74177.4 
92721.  7 
111266.0 

12930.4 
14777.6 

16624.  8 
18472.0 

36944-  o 
55416.0 
73887.  9 

92359.  9 
110831.  9 

12878.8 
14718.6 
16558.4 
18398.2 
36796.  5 
55194.7 
73592.  9 
91991.1 
110389.  4 

12826.  I 

14658.  5 
16490.  8 
18323.  i 
36646.  i 

54969.  2 

73292.  3 
91615-3 

109938.4 

12772.6 

H597-  2 

16421  9 
18246.  5 
36493.  o 
54739-  6 
72986.  i 
91232.  6 
109479.  i 

Meridional  Arcs  —  Values  of  Dm  ///  Yards. 

L.  24°  o'. 

I,  25°  o'. 

L.  26°  o'. 

235-5 

"  2!5'  5 

235-6 

8 

269.  i 

269.2 

269.  2 

O 

302.8 

302.8 

302.9 

10 

336.4 

336.5 

336.5         ' 

20 

672.9 

673.0 

673-1 

3° 

1009.3 

1009.4 

1009.  6 

40 
50 

1345.  7 

1682.  2 

1345-9 
1682.4 

1346.  i 
1  682.  (5 

60 

2018.6 

2018.  9 

" 

2019.  2 

7  oo 
8  oo 
9  oo 

14130.3 

16148.  9 
18167.5 

14132.  i 
16151.0 
18169.9 

I4I34.  I 
I6I53.2 

18172.4 

10  oo 

20186.  i 

20188.8 

20I9I.5 

20   00 

•     30  oo 
40  oo 
50  oo 
60  oo 

40372.  2 

60558.  3 
80744.  4 
100930.  5 
121116.  6 

40377-  5 
60566.  3 

80755-  i 
100943.  9 
121132.6 

• 

40383.  o 
60574.  6 
80766.  i 
100957.6 
121149.  i 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point.       . 

PROJECTION    OF    MAPS. 


I29 


Arcs  of  Parallel —  Values  of  T)p  in  Yards. 


L.  26°  30', 

L.  27°  o'. 

L.  27°  30'. 

L.  28°  o'. 

L.  28°  30'. 

L.  29°  o  , 

7 

212.0 

211.  O 

2IO.  I 

205.  i 

208.2 

207.2 

8 

242.2 

241  .2 

240.  I 

239-0 

237.9 

236.8 

9 

272.5 

271.3 

270.1 

268.9 

267.6 

266.4 

10 

302.8 

301.5 

30O.  I 

298.8 

297.4 

296.0 

20 

605.6 

603.0 

6OO.3 

597-6 

594-8 

591.9 

30 

908.4 

904.5 

900.4 

896.3 

892.2 

887.9 

40 

I2II.2 

1206.0 

I2OO.6 

1195.1 

1189.5 

1183.9 

50 

I5I4.0 

1507.4 

1500.7 

1493-9 

1486.9 

T479-9 

Co 

1816.9 

1808.9 

1800.9 

1792.7 

1784-3 

1775-8 

7  oo 

I27I8.0 

12662.5 

12606.0 

12548.6 

12490.  i 

12430.8 

8  oo 

14534-9 

M47I.4 

14406.9 

14341.2 

M274.5 

14200.6 

9  oo 

I635L7 

16280.3 

16207.7 

16133-9 

16058.8 

15982.5 

10  00 

18168.  6 

18089.3 

18008.6 

17926.5 

17843-1 

17758.3 

20  00 

36337.2 

36178.5 

36017.1 

35S53-0 

35686.2 

35516.6 

30  oo 

54505.3 

54267.8 

54025.7 

53779-5 

5352Q.3 

53274.9 

40  oo 

72674.4 

72357.1 

72034.3 

71706.  i 

71372.4 

71033.2 

50  oo 

90843.0 

90446  .  3 

90042  .  9 

89632.6 

89215.4 

88791.5 

60  oo 

109011.  5 

108535.6 

108051.4 

107559.1 

107058.5 

106549.8 

Meridional  Arcs — Values  of  D.*  in  Yards. 


L.  27°  o'. 

L.  28°  o'. 

L.  29°  o'. 

7 

235.6 

235-6 

235-7 

8 

269.3 

269.3 

269.3 

9 

302.9 

303-0 

303  .  o 

10 

336.6 

336.6 

336.7 

20 

673-1 

673.2 

673-3 

30 

]  009  .  7 

1009.9 

IOIO.O 

40 

T346.3 

T346.5 

1346.7 

50 

1682.9 

1683.1 

1683.3 

60 

2019.4 

2019.7 

2O2O.O 

7  oo 

14136.0 

14138.1 

I4I40.I 

8  oo 

16155-5 

16157.8 

I6I60.2 

9  oo 

18174.9 

18177-5 

l8l80.2 

10  oo 

20194.3 

20197.2 

2O2OO.  2 

20  oo 

40388.7 

40394.5 

40400.4 

30  oo 

60583.0 

60^91.  7 

60600  .  6 

40  oo 

80777.4 

80788.9 

80800.8 

50  oo 

100971.7 

100986.2 

IOIOOI  .O 

60  oo 

121166.0 

121183.4 

I2I2OI  .2 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

9 


130 


GEODESY. 


Arcs  of  Parallel —  Values  of  DI}  in  Yards. 


L.  29°  30'. 

L.  30°  o'. 

L.  30°  30'. 

L.  31°  o'. 

L.  31°  30'. 

L.  32°  o'. 

7 

206.2 

205.2 

204.1 

203.1 

202.  O 

200.9 

8 

235-6 

234.5 

233-3 

232.1 

230.9 

229.6 

9 

265.1 

263.8 

262.4 

201  .  I 

259.7 

258.3 

10 

294.5 

293.1 

291.6 

290.  I 

288.6 

287.0 

20 

589.1 

586.2 

^83.2 

580.2 

577-1 

574-0 

30 

883.6 

879.2 

874.8 

870.3 

865.7 

861.1 

40 

1178.1 

1172.3 

1166.4 

TI60.4 

II54-3 

1148.  i 

50 

1472.7 

1465.4 

1458.0 

1450.5 

1442.9 

I435-I 

60 

1767.2 

,  1750.5 

1749.6 

I74O.6 

I73L4 

1722.1 

7  oo 

12370.5 

12309.3 

12247.1 

I2IS4.0 

I2I20.0 

12055.0 

8  oo 

I4I37.7 

14067.7 

13996.7 

13924.6 

I385L4 

I3777-I 

Q  oo 

15904.9 

15826.2 

15746.3 

I5665.I 

15582.8 

15499-3 

10  00 

17672.2 

1/584.7 

17495.9 

17405.7 

I/3I4.2 

17221.4 

20  oo 

35344-3 

35169.4 

3499^-7 

348II.4 

34628.4 

34442.9 

30  oo 

53016.5 

52754.0 

52487-6 

522I7.I 

51942.7  i  51664.3 

40  oo 

70688.7 

70338.7 

69983.4 

69622.8 

69256.9 

68885.7 

50  oo 

88360.8 

87923.4 

S7479-3 

87028.5 

86571.1 

86107.  i 

60  oo 

106033.0 

10550*.  i 

104975.2 

104434.2 

103885.3 

103328.6 

Meridional  Arcs —  Values  of  Dm  in  Yards. 


L.  30°  o'. 

L.  31°  o'. 

L.  32°  o'. 

7 

235-7 

235-7 

235-8 

8 

269.4 

269.4 

209.5 

9 

303  .  o 

303-1 

303-1 

10 

336.7 

336.8 

"36.8 

20 

673-4 

673.5 

673.6 

30 

IOI0.2 

1010.3 

1010.  5 

40 

1346.9 

I347-I 

1347-3 

50 

1683.6 

1683.9 

1684.1 

60 

2020.3 

2O2O.6 

2020.9 

7  oo 

I4U2.3 

14144.4 

14146.6 

8  oo 

I6I62.6 

16165.  ! 

16167.6 

9  oo 

18182.9 

18185.7 

18188.5 

10  oo 

2O2O3.2 

2O2O6.3 

20209.5 

20   00 

40406  .  5 

40412.6 

40418.9 

30  oo 

60609.7 

60619.0 

60628.4 

40  oo 

80812.9 

80825.3 

80837.9 

50    00 

101016.1 

IOIO3I.6 

101047.4 

60  oo 

121219.4 

121237.9 

I  I2I256.8 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

PROJECTION    OF    MAPS.                                           131 

Arcs  of  Parallel  —  Values  of  D2>  in  Yards. 

L.  32.°  30'. 

L.  33°  o'. 

L.  33°  30'. 

L.  34°  o'. 

L.  34°  30'. 

L.  35°  o'. 

7 

199.8 

198.7 

197.6 

196.4 

195-3 

194.1 

8 

228.4 

227.1 

225.8 

224.5 

223.2 

221.8 

9 

256.9 

254-5 

254.0 

252.6 

251.1 

249.6 

10 

285.5 

283.9 

282.3 

280.6 

279.0 

277-3 

20 

570-9 

567.7 

564.5 

561  .2 

557-9 

554-k 

30 

856.4 

851.6 

846.8 

841.9 

836.9 

831.9 

40 

II4I.8 

IT35-5 

1129.0 

II22.5 

"15.9 

1109.2 

50 

1427.3 

I4I9.3 

I4H.3 

I403.I 

1394.8 

1386,4 

60 

1712.7 

1703.2 

1693-5 

1683.7 

1673-8 

1663.7 

7  oo 

11989.1 

11922.3 

11854.6 

II7S6.0 

11716.5 

11646.1 

8  oo 

13701.9 

13625.  .5 

I3548.I 

13469.7 

13390.3 

13309.8 

9  oo 

I54I4.6 

15328.7 

15241.7 

I5I53.5 

15064.1 

14973.6 

10  oo 

17127.3 

17031.9 

16935-2 

16837.2 

16737.9 

16637.3 

20   00 

34254.6 

34063.8 

33870.4 

33674.3 

33475-8 

33274.6 

30  oo 

51381.9 

51095-7 

50805.5 

505H.5 

50213.6 

49911.9 

40  oo 

68509.3 

68127.6 

67740.7 

67348.7 

66951.5 

66549.2 

50  oo 

85636.6 

85I59-5 

84675.9 

84185.8 

83689.4 

83186.5 

60  oo 

102763.9 

102191.4      101611.  i 

IOIO23.O 

100427.3 

99823-8 

Meridional  Arcs  —  Values  of  Dm  ///  Yards. 

L.  33°  o'. 

L.  34°  o'. 

L.  35°  o'. 

7 

235-8 

235'-  9 

235-9 

8 

269.5 

269.5 

269.6 

9 

303-2 

303-2 

303  •  3 

10 

336.9 

336.9 

337-0 

20 

673-8 

673-9 

6/4-0 

30 

IOIO.6 

1010.8 

ion  .0 

40 

1347-5 

1347.7 

1347-9 

50 

1684.4 

1684.7 

1684.9 

60 

2021.3 

2021.6 

202  i  .  9 

7  oo 

14148.9 

I4I5I.2 

I4I53.5 

8  oo 

16170.  i 

16172.7 

16175.  a 

g  oo 

18191.4 

18194.3 

18197-3 

10   00 

20212.7 

20215.9 

20219.2 

20  oo 

40425.4 

40431.9 

40438.5 

30  oo 

60638.0 

60647.8 

60657.7 

40  oo 

80850.7 

80863.7 

80877.0 

•      50  oo 
60  oo 

101063.4 
121276.1 

101079.7 
121295.6 

101096.2 
121315.4 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

132 


GEODESY. 


Arcs  of  Parallel —  Values  of  D ,  in  Yards. 


L.  35°  30'. 

L.  36°  o'. 

L.  36°  30'.  L.  37°  o'. 

L.  37°  30. 

L.  38°  o'. 

/   ;i 

| 

7 

192.9 

191.7 

190.5     189.3 

188.0 

186.8 

8 

220.5 

219.1 

217.7     216.3     214.9 

213.4 

9 

248.0 

246.5 

244.9     243.3 

241.7 

240.1 

10 

275.6 

273-9 

272.1     270.4 

268.6 

266.8 

20 

551.2 

547-7 

544.3     540.7 

537-2 

533-6 

30 

826.8 

821.6 

816.4  i   sii.i 

805.8 

800.4 

40 

1102.4 

1095.5 

1088.5    1081.5 

1074.4 

1067.2 

50 

1378.0 

1369.4 

1360.7    1351-9 

1343-0 

1334.0 

60 

1653.5 

1643.2 

1632.8  i   1622.2 

1611.6 

1600.7 

7  oo 

11574.8 

11502.7 

11429.6  ;  II355-7 

11280.9 

11205.2 

8  oo 

13228.4 

I3I45.9 

13062.4   12977.9 

12892.5 

12806.0 

9  oo 

14881.9 

14789.1 

14695.2   14600.2 

14504.0 

14406.7 

10  oo 

16535.5 

16432.4 

16328.0   16222.4 

16115.6 

16007.5 

20  oo 

33070.9 

32864.7 

32656.0  :  32444.8 

32231.1 

32015.0 

30  oo 

49606.4   49297.1 

48984.0   48667.2 

48346.7 

48022.5 

40  oo 

66141.9 

65729-5 

65312.1   64889.6 

64462.3 

64030.0 

50  oo 

82677.3 

82161.8 

81640.1  ,  81112.0 

'80577.8 

80037.5 

60  oo 

99212.8 

98594.2 

97968.1   97334-4 

96693.4 

96045.0 

1 

Meridional  Arcs —  Values  of  DJrt  in  Yards. 


L.  36°  o'. 

L.  37°  o'. 

L.  38°  o'. 

7 

235-9 

236.0 

236.0 

8 

269.6 

269.7 

269.7 

9 

303.3 

303-4 

303-4 

10 

337-0 

337-1 

337-2 

20 

674.1 

674.2 

674-3 

30 

IOII.  I 

ion  .3 

1011.5 

40 

1348.2 

1348.4 

1348.6 

50 

1685.2 

1685.5 

1685.8 

60 

7  oo 
8  oo 

2022^1 

14155^^ 
16178.1 

2022.6 
I4I5S.2 

16180.  8 

2022.9 

14160.6 
16183.5 

9  oo 

18200.3 

18203.4 

18206.5 

10   00 

2O222.6 

20226.0 

20229.4 

20  oo 

40445.2 

40451.9 

40458.8 

30  oo 

60667.5 

60677.9 

60688.2 

40  oo 

80890.3 

80903.9 

80917.6 

50  oo 

IOIII2.9 

101129.9 

101147.0 

60  oo 

I2I335-5 

I2I355.8 

121376.4 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

PROJECTION    OF    MAPS.                                              133 

Arcs  of  Parallel  —  Values  of  D7,  in    Yards. 

L.  38°  30'. 

L.39°  o'. 

L.3QC  30'. 

L.  40"  o'. 

L.40°  30'. 

L.4i°o'. 

i    a 
7           185-5 

184.2            182.9 

181.6 

180.2  |        178.9 

8               212.  O 

210.5  i         209.0 

207.5 

206.0  i        204.4 

9           238.5 

236.8           235.1 

233-4 

231.7          230.0 

10           265.0 

263.1            261.3 

259-4 

257.5          255.6 

20    |            529.9 

526.3           522.5 

518.8 

515-0          511-1 

30  ;      794.9 

789.4            783-8            77S.2 

772.4 

766.7 

40             1059.9 

1052.5 

1045.1          1037.5 

1029.9 

IO22.  2 

50    i          1324.9 

1315-6 

1306.3          1296.9 

1287.4 

1277.8 

60 

1589.8 

1578.8 

1567.6          1556.3 

1544.9        1533-4 

7  oo  '     11128.7 

11051.4 

10973.2  !     10894.1 

10814.3      10733.6 

S  oo  ;     12718.6 

12630.2 

12540.8  j     12450.4 

12359.2      12266.9 

9  oo        14308.4 

14208.9 

14108.4       14006.8 

13904.1       13800.3 

10  oo       15898.2 

15787-7- 

15676.0       15563-1 

15448.9 

15333-4 

20  oo       31796.4 

31575-4  !    31351.9  i    31126.1 

30897.9 

30667.3 

30  oo       47694.6 

47363.1      47027.9      46689.2 

46346.8 

46OOI.O 

40  oo  j     63592.8 

63150.8      62703.9  !    62252.2 

61795-8 

'61334.6 

50  oo       79491.0 

78938.4      783/9-9  :<    77815-3 

77244.7 

76668.3 

60  oo       95389.2 

94726.1 

94055.8 

93378.3 

92693.7 

92OO1.9 

Meridional  Arcs  —  Values  of  Dm  in    Yards. 

L.  39°  o'. 

L.  40°  o'. 

L.  41°  o'. 

1 

236.0 

236:1 

236.1 

8 

269.8 

269.8 

269.9 

9 

303-5 

303-5 

303.6 

10 

337-2 

337-3 

337-3 

20 

674.4 

674.5 

674.7 

30 

ion  .6 

ion  .8 

IOI2.0 

40 

1348-9 

I349-I 

1349-3 

50 

1686.1 

1686.4 

1686.7 

60 

2023.3 

2023.6 

2024.0 

7  oo 

14163.0 

14165.4 

14167.9 

8  oo 

16186.3 

16189.1 

16191.9 

9  oo 

18209.6 

18212.7 

I82I5.8 

10   00 

20232.8 

20236.3 

20239.8 

20  oo 

40465  .  7 

40472  .  7 

40479.7 

30  oo 

60698.5 

60709  .  o 

60719.5 

40  oo 

80931.4 

80945-3 

80959.3 

50  oo 

101164.2 

101181.6 

IOI199.2 

60  oo 

121397-1 

121418.0 

I2I439.0 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

J34                                                          GEODESY. 

Arcs  of  Parallel  —  Values  of  Dp  in    Yards. 

L.4I°  30'. 

L.  42°  o'. 

L.42°  30'. 

L.  43=  o'. 

L-43°  30'. 

L.  44°  o'. 

7 

177-5 

176.2 

174-8, 

173-4 

172.0 

170.5 

8 
9 

202.9 

228.3 

201.3 
226.5 

199.7 
224.7 

198.1            196.5 

222.9    •            221.  I 

194.9 
219.3 

10 

253-6 

251-7 

249.7 

247-7    j            245.7 

243-6 

20 

507.2 

503.3 

449-4 

495-4           49L3 

487-3 

30 

760.9 

755-0 

749.0 

743.0  i         737.0 

730.9 

40 

50 

1014.5 
I268.I 

1006.  6 
1258.3 

998.7 
1248.4 

990.7           982.7 
1238.4          1228.3 

974-5 
1218.1 

60 

1521.7 

1510.0 

1498.1 

1486.1          1474.0 

1461.8 

7  oo 

10652.0 

10569.7 

10486.6 

10402.6       10317.9 

10232.3 

8  oo 
9  oo 

I2I73-3 
13695.5 

12079.7 
13589-6 

11984.6 
13482.7- 

11888.7 
13374-8 

11791.8 
13265.8 

11694.  i 
I3I55.S 

10   00 

I52I7.2 

15099.6 

14980.8 

14860.9 

14739-8 

14617.6 

20  oo 
30  oo 
40  oo 

30434.4 
45651.6 
'    60868.8 

30199.1 
45298.7 
60398.3 

29961.6 
44942.4 
59923.2 

29721.7 
44582.6 
59443-4 

29479.6 
44219.4 

58959-2 

29235.2 
43852.8 
58470.4 

50  oo 
60  oo 

76086.0 
91303.2 

75497-9 
90597.4 

74903.9 

89884.7 

74304.3 
89165.1 

73698.9 

88438.7 

73088.0 
87705.6 

Meridional  Arcs  —  Values  of  Dw  in    Yards. 

L.  42°  o'. 

L.  43°  o'. 

L.  44°  o'. 

7 

236.2 

236.2 

236.3 

8 

269.9 

270.0 

270.0 

9 

303.7 

303.7 

303.8 

10 

337-4 

337.4 

337-5 

20 

674.8 

674.9 

675.0 

30 

IOI2.2 

1012.3 

1012.5 

40 

1349.6 

1349.8 

1350.0 

50 

1686.9 

1687.2 

1687.5 

60 

2024.3 

2024.7 

2025.0     . 

7  oo 

I4I70.3 

14172.8 

I4I75.3 

8  oo 

16194.7 

16197.5 

16200.3 

9  oo 

18219.0 

18222.2 

18225.4 

10  oo 

20243.4 

20246.9 

20250.4 

20   00 

40486.7 

40493  •  8 

40500.9 

30  oo 

60730.1 

60740.7 

60751.3 

40  oo 

80973.4 

80987.5 

81001.7 

50  oo 

IOI2I6.8 

101234.4 

101252.2 

60  oo 

I2I460.I 

1 

121481.3 

121502.6 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

PROJECTION    OF    MAPS. 


Arcs  of  Parallel —  Values  of  Dp  in    Yards. 


L.44°  30'. 

L.45°o'. 

L.45°30'. 

L.  46°  o'. 

L.46°  30'. 

L.  47°  o'. 

7 

i6g.i 

167.6 

166.2 

164.7 

163.2 

161  .7 

8 

IQ3-3 

191.6 

189.9 

188.2     186.5 

184.8 

9 

217.4 

2i5o 

213.7 

211.  8  i    209.8 

207.9 

10 

241.6 

239-5 

237-4 

235.3   233.2 

331-0 

20 

483.1 

479.0 

474-8 

470.6  j   466.3 

.  462.0 

30 

724.7 

7i3.5 

712.2 

705.9  !    699-5 

693-1 

40 

966.3 

958.0 

949-6 

941.2  !    932.7  1   924.1 

50 

1207.9 

H97.5 

1187.0 

1176.5  I   1165.8 

"55-1 

Go 

1449.4 

1437-0 

1424-4 

1411.8    1399-° 

1386.1 

7  oo 

10146.0 

10058.9 

9971-0 

gS32.4  |   9793-0 

9702.8 

8  oo 

1  1  595  -'4 

11495.9 

II395-5 

1129.1.2 

11192.0 

11089.  c 

g  oo 

13044.8 

12932.9 

12819.9 

12705.9 

12591.0 

12475.1 

10  00 

14494.3 

14369.8 

14244.3 

14117.7 

13990-0 

13861  .2 

20  oo 

28988.6 

28739.7 

28488.6 

28235.4   27980.0 

27/22.4 

30  oo 
40  oo 
50  oo 
60  oo 

43482.8 
57977-1 
72471.4 
86965.7 

43109.5 

57479-4 
71849.2 
86219.1 

42733.0 
56977.3 
71221.6 
85465.9 

42353.I 
56470.8 

70588.5 
84706.2 

4ig7O.o 
55g6o.o 
69950.0 
83940.0 

41583.  C 
55444-5 
69306.0 
83167.: 

Meridional  Arcs  —  Values  of  Dm  in    Yards. 

\ 

L.  45°  o'. 

L.  46"  o'. 

L.  47°  o'. 

'"' 

236.3 

236.3 

236.4 

8 

270.  1 

270.  i 

270.1 

9 

303.8 

303.9 

303-9 

10 

337-6 

337-6 

337-7 

20 

675.T 

675o 

675.4 

30 

1012.7 

1012.9 

1013.1 

40 
1                   50 

1350.3 

1687.8 

1350.5 
1688.1 

1350.7 
1688.4 

60 

2025.4 

2025.8 

2026.1 

7  oo 

14177.8 

14180.3 

14182.8 

8  oo 

16203.2 

, 

16206.0 

16208.  g 

g  oo 

18228.6 

18231.8 

18235.0 

10   00 

20254.0 

20257.5 

20261  .  i 

| 

20  oo 

40508.0 

40515.1 

40522.2 

30  oo 

60761.9 

60772.6 

60783.2 

40  oo 

81015.9 

81030.1 

81044.3 

50  oo 

101269.9 

101287.7 

101305.4 

60  oo 

121523.9 

121545.2 

121566.5 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal  point. 

I36                                                          GEODESY. 

Arcs  of  Parallel  —  Values  of  Dp  in    Yards. 

L.47°  30'. 

L.  48°  o'. 

L.48°  30'.  :  L.  49°  o'.  1  L.  49°  30'. 

L.  50°  o'. 

| 

7 

160.2 

158.7           157-1  ,         155-6           154-0 

152.4 

8 

183.1 

181.3           179-6  i         177.8           176.0 

174.2 

9 

206.0 

204.0           202.  o 

20O.O                igS.O 

196.0 

10 

228.9 

226.7           224.5 

222.3                220.0 

217.8 

20 

457-7 

453-3            449-0 

444-5  !         440.1 

435-6 

30 

686.6 

6So.o           673.4 

666.8            660.  i 

653.3 

40 

9J5-4 

906.7 

897.9 

889.0            880.1 

871.1 

50 

H44.3 

"33-4 

1122.4 

mi.  3  i       noo.i 

1088.9 

60 

I373-I 

1360.0 

1346.9 

1333-6 

1320.2 

1306.7 

1 

7    GO 

8  oo 

9612.0 

10985.1 

9520.3          9428.0 
10880.4  i     10774.8 

9334-9 
10668.5 

9241.1 
10561.2 

9146.6 
10453.2 

9  oo 

12358.2 

12240.4 

12121.7 

I2OO2.O 

11881.4 

H759-9 

IO   OO 

'I373I.4 

13600.5 

13468.5 

13335.6 

13201.6 

13066.5 

20  oo 

27462.7 

27200.9 

26937.1 

26671.  I 

26403.1 

26133.  r 

30  oo 

41194.1 

40801  .4 

40405.6 

40006  .  7 

39604.7 

39199.6 

40  oo 

54925.5 

54401.9 

53874.1 

53342.3 

52806.2 

52266.2 

50  oo 
60  oo 

68656.8 

82388.2 

68002  .  4 
81602.8 

67342.7 

808  1  1  .  2 

66677.8 
80013.4 

66007.8 
79209.4 

65332.7 
78399.3 

Meridional  Arcs  —  Values  of  DCT  in   Yards. 

L.  48°  o'. 

L.  49°  o'. 

L.  50°  o'. 

7 

236.4 

236-5 

236.5 

8 

2-0.2 

270.2 

270.3 

g 

304.0 

304.0 

304.1 

10 

337-7 

337-8 

337-9 

20 

675.5 

675.6 

675.7 

30 

IOI3.2 

1013.4 

1013.6 

40 

I35I.O 

I35L2 

I35L4 

50 

1688.7 

1689.0 

1689.3 

60 

2026.5 

2026.8 

2027.2 

7  oo 

I4I85.2 

14187.7 

14190.2 

8  oo 

I62II.7 

16214.5 

16217.3 

9  oo 

18238.2 

18241.3 

18244-5 

10   00 

• 

20264.6 

20268.1 

20271.7 

20  oo 

40529.2 

40536-3 

40543.3 

30  oo 

60793.9 

60804.4 

60815.0 

40  oo 

81058.5 

81072.6 

81086.6 

50  oo 

IOI323.I 

101340.7 

101358.9 

60  oo 

121587.7 

121608.9 

121629  9 

Intermediate  minutes  and  seconds  will  be  found  by  moving  the  decimal-point. 

PROJECTION    OF    MAPS.                                             137 

Lengths  in  Nautical  Miles  and  Statute  Miles  of  Degrees  of  Latitude 

and  Longitude  in  Different  Latitudes. 

DEGREE   OF   THE    PARALLEL.                        DEGREE   OE   THE   MERIDIAN. 

Latitude  of 

Nautical 

Statute 

i  Latitude  of 

Nautical 

Statute 

Parallel. 

miles. 

miles. 

middlepoint 

miles. 

miles. 

20 

56.404 

65.018 

20 

59-664 

68.777 

21 

56.039 

64-598 

22 

55.657 

64-158 

23 

55.258 

63.698 

24 

54-843 

63-219 

25 

54-411 

62.721                 25 

59-706 

68.825 

26                     53.962 

62.204 

27               53-497 

61.668 

2S 

53-016 

61.113 

29 

52.518 

60.540 

30               52.005 

59.948 

30 

59-749 

68.875 

31                5L476 

59.338 

32                50.931 

58-709 

33               50.370 

58.063 

34               49-794 

57-399 

35               49-203 

56.718 

35 

59.796 

68.929 

36               48.597 

56.019 

37               47-976 

55.304 

33 

47-341 

54-571 

39               46.690 

53-822 

. 

40 

46.026 

53-056 

40 

59-847 

68.987 

41 

45.348 

52.274 

42 

44.654 

5L476 

43 

43-949 

50.662 

44 

43.230 

49-833     | 

45 

42.497 

48.988                 45 

59.899 

69.048 

46 

41.752 

48.128 

47 

40.993 

47-254 

48 

40.222 

46.365 

49 

39-439 

45-462 

50 

38.643 

44-545 

50 

59-951 

69.108 

A  degree  of  longitude  at  the  equator  =  69.163  statute  miles. 

A  second  of  time  at  the  equator          =  1521.6  feet. 

138 


GEODESY. 


Co-ordinates  of  Curvature  in  Statute  Miles  for  Maps  of  Large  Extent. 


i 

Latitude  20°. 

Latitude  22°. 

Latitude  24°. 

Latitude  26°. 

fc/3 

e 

0 

dn 

dp 

*,      1   4 

dm 

dp 

dm 

dp 

1-1 

1 

2 

130.0 

0.8 

I2S.3 

0.8 

126.4 

0.9 

124.4 

-    0.9 

4 

260.0 

3-i 

256.6 

3-3 

252.8 

3-6 

248.8 

3-8 

.     6 

390.0 

6.9 

384.8 

7-5 

379-2 

8.1 

373-1 

8.6 

8 

520.0 

12.4 

513.0 

13-4 

505.5 

14.4 

497-3 

15-2 

10 

649.8 

19.4 

64I.I 

21.0 

631-.  7 

22.4 

621.4 

23.8 

12 

779-7 

27.8 

769.1 

30.2 

757-9 

32.2 

745-4 

34-2 

14 

909.2 

38.0 

896.9 

41.0 

883.6 

43-9 

869.2 

46.6 

16 

1039.2 

49-6 

1024.5 

53-6 

1009.9 

57-4 

992.8 

60.8 

18 

1168.1 

62.8 

II52.2 

67-9 

1134-8 

72.6 

1116.  i 

77.0 

20 

1298.0 

77.6 

1279-5 

83.8 

I20I.2 

89.7 

1239.2 

95-0 

r 

10892 

9813 

'      8905 

8130 

Longitude. 

Latitude  28°. 

Latitude  30°. 

Latitude  32°. 

Latitude  34°. 

dm 

4 

dm 

4 

dm 

dp 

dm 

dP 

2 

122.2 

I.O 

119.8 

i  .0 

117.4 

i.i 

114.8 

i.i 

4 

244  .  4 

4.0 

239-7 

4.2 

234.8 

4-3 

229.5 

4-5 

6 

366.5 

9.0 

359-5 

9-4 

352.0 

•9.8 

344-2 

IO.  I 

8 

488.6 

16.0 

479.2 

16.7 

469-3 

17-3 

458.7 

17.9 

10 

610.4 

25.0 

598.7 

26.1 

586.3 

27.1 

573-1 

28.0 

12 

732.4 

36.0 

718.0 

37-6 

703.5 

39-1 

687.2 

40.3 

14 

853.7 

49.0 

837-1 

51-2 

819.6 

53-1 

801.1 

54-8 

16 

975-7 

64.1 

956.0 

66.9 

936.8 

69-5 

9M-7 

71.6 

18 

1096.0 

80.9 

1074.6 

84.6 

1051.9 

87.8 

1027.9. 

9°-5 

20 

1218.  8 

IOO.  I 

1192.9 

104.3 

1169.2 

108.6 

1140.7 

in.  7 

r 

7458 

6869 

6348 

5881 

PROJECTION    OF    MAPS. 


Co-ordinates  of  Curvature  in  Statute  Miles  for  Maps  of  Large  Extent. 


Longitude. 

Latitude  36°. 

Latitude  38°. 

Latitude  40°. 

Latitude  42°. 

dm 

dp 

dm 

4 

dm 

dp 

dm 

4 

2 

112.  O 

1.2 

109.1 

1.2 

1  06.  1 

1.2 

102.9 

1.2 

4 

224.0 

4.6 

218.2 

4-7 

212.2 

4.8 

205.8 

4.8 

6 

335-9 

10.3 

327.2 

10.6 

3I8.I 

10.7 

308.6 

10,8 

8 

447-7 

18.4 

436.0 

18.8 

423.9 

18.9 

4II.2 

19.2 

10 

559-2 

28.7 

544-7 

29-3 

529.4 

29-7 

5I3.6 

30.0 

12 

670.5 

41-3 

653.0 

42.2 

634.7 

42.8 

615.7 

43-2 

14 

781.6 

56.2 

761.  i 

57-4 

739-7      53.2 

717.5 

58.8 

16 

892.3 

73-4 

868.8 

74-9 

844.3      76.0 

818.8 

76.7 

18 

1002  .  6 

92.8 

976.2 

94-7 

948.5      96.1 

919.8 

97-0 

20 

1112.5 

II4-5 

1083.0  niG.S 

1052.3    118.5 

1020.2 

119.7 

r 

5461 

5079 

4729 

4408 

Longitude. 

Latitude  44°. 

Latitude  46°. 

Latitude  48°. 

Latitude  50°. 

.</„ 

dp 

dm 

4 

«,    j  * 

d» 

0 

2 

99-7 

1.2 

96.2 

1.2 

92.7          1.2 

89.1 

1.2 

4 
6 

198.9 
298.7 

4.8 
10.9 

192.4 

288.5 

4.8 
10.9 

185.4  !     4-8 
277.9     IO-8 

178.1 
267.0 

4.8 
10.7 

8 

398.0 

19-3 

384-4 

19.3 

370.3     19-2 

355-7 

19.0 

.  10 

497-1 

30.2 

480.0 

30.2 

462.3  !  30.0 

444-1 

29-7 

12 

595-9 

43-4 

575-4 

43-4 

554-1   j  43-2 

532-3 

42.8 

14 

694-3 

59-1 

670.3 

59-1 

645.6      58.8 

620.0 

58.2 

16 

18 

792.3 
889.9 

77-1 
97-5 

764.9 
859.0 

77-1 

97-5 

736.5      76.7 
827.0     97.0 

707.3 
794.1 

75-9 
96.0 

20 

986.9 

1  2O.  2 

952.5 

I2O.2 

916.9    119.6 

880.3 

118.4 

r 

4110 

3833 

3575 

3332 

140  GEODESY. 


X  L I V . — Trigonometrical  Leveling . 

Reciprocal  zenith-distances  measured  at  two  stations  at  the 
same  time  give  the  best  results.  When  reciprocal,  but  not  simul 
taneous,  the  observations  should  be  made  on  different  days,  in 
order  to  obtain  as  far  as  possible  an  average  value  of  the  refrac 
tion  as  well  as  the  mean  value  of  the  difference  between  the 
respective  angles;  the  same  with  zenith-distances  measured  at 
one  station  only. 

The  refraction  being  greater  and  more  variable  at  sunrise  and 
sunset,  and  comparatively  stationary  between  the  hours  of  10 
a.  m.  and  4  p.  m.,  the  best  time  for  observation  is  between  those 
hours. 

The  condition  of  the  atmosphere  and  the  relative  refraction 
may  be  so  different  at  stations  more  than  twenty  miles  apart,  that, 
as  a  general  rule,  the  difference  of  level,  determined  even  by 
reciprocal  observations,  cannot  be  relied  upon  for  much  accuracy 
at  greater  distances  unless  a  very  large  number  of  measurements 
have  been  made  under  the  most  favorable  circumstances.  The 
higher  the  elevations  the  more  reliable  the  results. 


i. — To  obtain  the  Difference  of  Level  of  Two  Points  from  Reciprocal 
Zenith- Distances,  simultaneous  or  not. 

Let— 

Z,  Z'  =  the  measured  zenith-distances  of  the  telescopes  at 
the  two  stations,  of  which  Z  is  the  smaller ; 

K  =  distance  in  yards  between  the  tw6  stations ; 

R^=  radius  of   curvature    of    the    arc   joining  the   two 
stations  ; 

C  =  angle  at  the  earth's  center  subtended  by  the  arc ;  and 
dh  =  difference  of  level  of  the  two  stations ; 


then — 


K  ji      K  sin  fr  (Z'  -  Z) 

=  K7*n~7"  ;  d~  =  cos  J  (Z'" -  Z  +  C) 


TRIGONOMETRICAL  MEASUREMENT  OF  HEIGHTS. 


141 


XLIV. — Trigonometrical  Leveling — Continued. 

If  the  telescope  is  not  observed  upon,  but  some  other  object 
near,  the  measured  zenith-distance  can  be  reduced  to  the  tele 
scope  by  the  formula : 

Reduction  to  telescope,  in  seconds,  =  i  - 

K  sin  i" 

in  which  r  represents  the  distance  of  the  object  (in   yards   and 
decimals)  above  or  below  the  telescope. 

Logarithmic  values  of  R2,  in  yards,  which  depend  on  the  inclina 
tion  to  the  meridian  of  the  arc  joining  the  two  stations  and  their 
mean  latitude,  are  given  in  the  following  table : 


I 

i>  •-£ 

Lat.  25°. 

Lat.  30°. 

Lat.  35°. 

Lat.  40". 

Lat.  45°. 

Lat.  50°. 

^ 

1  ~" 

logR, 

"** 

log  Rr 

log  R, 

logR* 

log  R.Z 

o 

6.841384 

6.841695 

6.842039 

6.842406 

6.842785 

6.843164 

10 

M56 

1760 

2098 

2458      2829 

3200 

20 

1663 

1950 

2267 

2606 

2955 

3304 

30 

1981 

2240 

2527 

2833 

3148 

3464 

40 

2370 

2596 

2845 

3HI 

3386 

3631 

50 

2785 

2975 

3184 

3408 

3639 

3870 

bo 

3176 

3331 

3504 

3687 

3877 

4066 

70 

3494 

3622 

3764 

3915 

4069 

4227 

80 

3702 

3812 

3934 

4063     4197 

4331 

90 

6.843774 

6.843878 

6.843993 

6.844115 

6.844241 

6.844368 

log  sin  i"  =  4.685575 

2. — By  the  Zenith- Distance  measured  at  one  Station. 
Let— 

Z  =  the  measured  zenith-distance  of  the  signal  or  object; 
K  =  the  distance  between  the  two  stations  in  yards; 
m  =  the  co-efficient  of  refraction  =  0.071 ';  and 
dh  =  difference  in  height  between  the  two  stations ; 
then — 


c  = 


K 


sn 


dh  —  K  cos  (Z  +  m  C  ~  i_ 
sin  (Z  +  m~C  —  C} 


I42  GEODESY. 


XLI V. — Trigonometrical  Leveling — Continued. 

3- — By  the  Observed  Zenith-Distance  of  the  Sea-Horizon. 
Let— 

Z  =  the  measured  zenith-distance ; 
R^  =  the  radius  of  curvature  of  the  arc ;   and 
m  —  the  co-efficient  of  refraction  =  0.078 ; 
then — 


=  -  -,,  tan2  (Z  -  90°) 

21  —  m\l 


h 


4- — By  Observed  Angles  of  Elevation  or  Depression. 
Let— 

A  =  the  observed  angle  expressed  in  seconds ;  and 
K  ==  the  distance  in  yards  between  the  two  stations. 
dJi  =.  0.00000485  K  A  i  0.000000667  K-2 
[log  4.68574]  [log  2.8241^] 

This  gives  the  difference  in  heights  between  stations  not  more 
than  ten  or  fifteen  miles  apart,  with  a  probable  error  less  than 
the  uncertainty  in  the  co-efficient  of  refraction. 

Co-efficient  of  Refraction. 

The  co-efficient  of  refraction,  or  proportion  of  the  intercepted 
arc,  is  determined  from  the  observed  zenith-distances  of  two  sta 
tions,  the  relative  altitudes  of  which  have  been  determined  by 
the  spirit-level ;  or  from  reciprocal  zenith-distances,  simultaneous 
or  not,  under  the  assumption  that  the  mean  of  a  number  of  ob 
servations  taken  under  favorable  conditions  will  eliminate  the 
difference  of  refraction  which  is  found  to  exist,  even  at  the  same 
moment  at  two  stations  a  few  miles  apart. 

The  longer  the  distance  the  greater  is  the  error  caused  by  any 
uncertainty  in  the  co-efficient,  or  in  the  actual  refraction;  conse 
quently  there  is  a  limit  to  the  distance  for  which  any  assumed  mean 
value  of  the  refraction  can  be  depended  on  for  accurate  results. 


TRIGONOMETRICAL  MEASUREMENT  OF    HEIGHTS.  143 

X  L  I  V.  —  Trigonometrical  Leveling  —  Continued. 

The  average  value  of  the  co-efficient  from  the   Coast   Survey 
observations  in  the  New  England  States  — 

Between  primary  stations  =  0.071 
Of  small  elevations.  .....  =0.075 

Of  the  sea-horizon  ......  =0.078 

In  the  trigonometrical  survey  of  Massachusetts  Mr.  Borden 
used  0.0784  as  a  mean  co-efficient  for  the  sea-co'ast,  and  0.0697 
for  the  interior  of  the  State. 

1.  —  To  determine  the   Co-efficient   of  Refraction  from  Reciprocal 

Zenith-  Distances. 

Let- 

C  =  angle  at  earth's  center  subtended  by  arc  ; 
F  =  angle  of  refraction  ;  and 
m  =  co-efficient  of  refraction  ; 

then— 

C  =  =-^--/7;     F=C_£(Z'  +  Z-  180°);       m  =  * 
R,  sini"  2       2  v  C 

2.  —  To  determine  the  Co  -efficient  from  the  Zenith-Distance  observed 

at  one  Station,  when  the  Altitude  of  the  Two  Stations  above  Tide, 
or  their  D  iff  ere  tic  e  in  Height,  have  been  determined  by  the  Leve  ling- 
Instrument. 

Compute  the  true  zenith-distances,  Z(/  and  Z0,  of  the  two 
given  points,  and  the  difference  between  the  true  and  the  observed 
zenith-distances  will  be  the  angle  of  refraction  F. 


J  (Z«'  +  Z0)  =  90 
*  (7  '       7  )  -        i-1         / 


Z0  —  Z  =  F;  m=~c 

h  and  h1  having  been  determined  by  the  leveling-instrument. 


144 


GEODESY. 


Difference  between  the  Apparent  and  the  7 rue  Level. 
Correction  for  curvature . . 

2R 

Correction  for  refraction =  I  -    m 

Correction  for  curvature  and  refraction  =  (i  —  2  w)  — 

'2R 

where — 

D=  the  distance; 

R  =  mean  radius  of  the  earth ;  and 
.      ;;/  =  co-efficient  of  refraction ; 

or — 

m  being  =  0.075,  an<*  log  R  in  feet  =  7.31991307, 

Correction  for  curvature,  in  feet,  =  log  D-  —  const  log  [7.6209430] 
Correction  for  refraction,  in  feet,  —log  D*  -f-  const  log  [1.5551483] 

Corrections  for  Curvature  and  Refraction,  showing  the  Differences 
of  the  Apparent  and  True  Level,  in  Feet  and  Decimals  of  a  Foot, 
for  Distances  in  Miles. 


en 

<u 

Difference  in  feet  for  — 

1 

Difference  in  feet  for— 

S 

g 

o 

e 

s 

Curva 
ture. 

Refraction. 

Curvature 
and 
refraction.. 

Distance, 

Curva 
ture. 

Refraction. 

Curvature 
and 
refraction. 

1 

0.7 

O.I 

0.6 

13 

112.  8 

16.9 

95-9 

2 

2-7 

0.4 

2-3 

14 

130.8 

19.6 

III.  2 

3 

4 

6.0 
10.6 

0.9 
1.6 

5-1 

9.0 

15 
16 

150.2 

170.8 

22.5 
25.6 

127.7 
145-2 

5 

16.7 

2.5 

14.2 

17 

192.9 

28.9 

164.1 

6 

24.0 

3-6 

20.4 

18 

216.2 

32.4 

183.8 

7 

32.7 

4.9 

27.8 

19 

240.9 

36.1 

204.8 

8 

42.7 

6.4 

36.3 

20 

266.9 

40.0 

226.9 

9 

10 

54-1 

66.7 

8.1 

10.  0 

44-0 

56.7 

21 
22 

294.3 
323.0 

44.1 

48.4 

25O.2 
274.6 

ii 

80.7 

12.  I 

68.6 

23 

353-0 

52.9 

300.1 

12 

96.1 

14.4 

81.7 

24 

384-4 

57-7 

326.7 

GEODESY.                                                          j^r 

Reduction,  in  Feet  and  Decimals,  upon  i  oo  Feet,  for  the  following 

Vertical  Angles. 

Angle. 

Reduc.      Angle. 

Reduc. 

Angle. 

Reduc. 

Angle. 

Reduc. 

0             / 

0            / 

0           / 

0           / 

3     o 

•137 

7  30 

.856 

•    12      O 

'    2.185 

16  30 

4.118 

3  15 

.161 

7  45 

.913 

12    15 

2.277 

16  45 

4-243 

3  30 

.187 

8     o 

•973 

12    30 

2.370 

17     o 

4-370 

3  45 

.214 

8  15 

1.035 

12  45 

2.466 

17  15 

4.498 

4     o 

.244 

8  30 

1.098 

13      0 

2-553 

17  30 

4.628 

4  15 

•  275 

8  45 

1.164 

13  15 

2.662 

17  45 

4.760 

4  30 

.308 

9     o 

1.231 

13  30 

2.763 

18     o 

4.894 

4  45 

•343          9  15 

1.300 

13  45 

2.866 

18  15 

5-030 

5     o 

.381    i       9  30 

i.37i 

14    o 

2.970 

1  8  30 

5-168 

5   15 

.420 

9  45 

•444 

14  15 

2.077 

18  45 

5-307 

5  30 

.460 

10      0 

.519 

14  30 

3-185 

19     o 

5.448 

5  45 
6     o 

.503         10  15 

.548  ;     10  30 

•596 
•  675 

14  45 

15      0 

3.295 
3-407 

19  15 
19  30 

5.591 
5.736 

6  15 

•594 

10  45 

•755 

15  15 

3.521 

19  45 

5.882 

6  30 
6  45 

•  643 
•693 

II      0 

ii  15 

.837 
.921 

15  30 
15  45 

3-637 
3.754 

20     o 

6.031 

7     o 

•745 

ii  30 

2.008 

16     o 

3.874 

7  15 

.800 

ii  45 

2.095 

16  15 

3-995 

Ratio  of  Slopes  for  the  following  Vertical  Angles. 

To  one 

To  one 

To  one 

To  one 

Angle. 

perpen 

Angle. 

perpen 

Angle. 

perpen 

Angle. 

perpen 

dicular. 

dicular. 

dicular. 

dicular. 

o         ; 

0            / 

0           / 

0            1 

o  15 

229 

3  35 

16 

8     8 

7 

18  26 

3 

o  30 

H5 

3  49 

15 

8  45 

J9  59 

2| 

o  45 

76 

4     6 

14 

9  27 

6 

21    48 

I       O 

57 

4  24 

13 

9  52 

5t 

23  =;8 

2-L 

i   15 

46 

4  45 

12 

10  18 

51 

26  34 

2 

i  30 

39 

5     o 

Irl 

10  47 

5i 

29  44 

i  45 

33 

5   12 

II 

ii  19 

5 

33  42 

jJL 

2      O 

28 

5  27 

iol 

ii  53 

4f 

38  40 

jl 

2    15 

25 

5  42 

IO 

12    32 

45     o 

I 

2    30 

23 

6    o 

91 

13    15 

4i 

53     8 

f 

2  45 

21 

6    21 

9 

14      2 

4 

63  28 

3     o 

19 

6  43 

84- 

14  55 

31  i 

75  58 

1 

3  15 

3  28 

18 
17 

7     7 
7  36 

8 

71 

15  56 
17     6 

31 

si  ; 

78  41 

.      i 

,- 

10 


146  GEODESY. 


XLV.  —  Barometrical  Measurement  of  Heights. 

TO    OBTAIN   THE    DIFFERENCE    IN   THE   HEIGHT  OF  TWO    PLACES  BY 
MEANS    OF    THE    BAROMETER. 

The  following  tables  have  been  condensed  from  those  in  the 
appendix  of  Lieutenant-Colonel  Williamson's  Treatise  on  the  Use 
of  the  Barometer,  etc.,  Professional  Papers,  Corps  of  Engineers, 
No.  15,  and  are  those  of  Plantamour  (Guyot's  tables  D,  72-79) 
re-arranged  and  adapted  to  English  measures. 

They  are  based  upon  Bessel's  formula,  which  differs  from  that 
of  La  Place  principally  in  containing  a  factor  depending  upon 
the  humidity  of  the  air.  The  modifications  introduced  by  Plan 
tamour  consist  in  some  slight  changes  in  the  values  of  the  baro 
metric  constants  in  accordance  with  the  supposed  more  accurate 
results  obtained  from  the  experiments  of  Regnault. 

La  Place's  formula  reduced  to  English  measures,  as  given  by 
Guyot,  (page  D,  35,)  is  : 


=  iog      x  60158.6  English  feet  ^  (i  +  0.00260  cos  2  L) 


\         CO 

9°°   / 


_     _          _ 

_    20886860  10443430 

Williamson  adopts  the  same  convenient  form  in  his  reduction 
of  Plantamour's  formula  to  English  measures  ;  thus  : 


982.2647; 

7  (1  +  0.0026257  cos  2  L) 

Z  =  logfi  X  60384.3  English  feet  <  7+52252 

~ 


s  __  \ 
3430^ 


where  h  and  H  are  the  heights  of  the  barometer  reduced  to 
32°  Fahrenheit;  /,  /',  the  temperatures  of  the  air  at  the  two  sta 
tions;  and  ;«,  a  factor  depending  upon  the  humidity  of  the  stratum 
of  air  between  them;  L  the  latitude  of  the  place. 


BAROMETRICAL   MEASUREMENT  OF  HEIGHTS. 


XLV. — Barometrical  Measurement,  &c. — Continued. 

APPLICATION. — i.  Reduce  the  readings  of  the  barometer  to 
32°  Fahrenheit  by  Table  I. 

2.  Representing  by  A  the  constant  60384.3,  Table  II  gives 
the  value  "of  A  log  h  or  A  log  H,  and  consequently  their  differ 
ence,  A  j— ^rf-'    The  numbers  have  been  diminished  by  a  con 
stant  quantity,  which  does  not  affect  their  differences. 

3.  Table  III,  column  B,  gives  values  of  the  factor  ^+^—64 

982.2647 

of  the  temperature-term,  to  be  used  only  in  connection  with  the 
tables  that  give  the  corrections  for  humidity. 

Column  C  gives  values  of  the  factor  _i ~°4    and  is  used 

900 

where  no  observations  of  atmospheric  humidity  are  made. 

4.  Table  IV  gives  A  log  ~  x  0.0026257  cos  2  L>  the  correc- 

rl 

tion  due  to  gravity  at  the  sea-level  in  the  mean  latitude  between 
the  two  stations.  It  is  positive  from  45°  to  the  equator,  and 
negative  from  45°  to  the  poles. 

5.  Table  V  shows  the  correction  A  log  h  x^"^2-5-  to  be 

H       20886860 

added  to  the  approximate  difference  of  altitude,  on  account  of 
the  decrease  of  gravity  on  a  vertical  acting  on  the  density  of  the 
mercurial  column. 


6.  Table  VI  furnishes  the  small  correction  A  loir  - 


x 


H      10443430 

for  the  decrease  of  gravity  on  a  vertical  acting  on  the  density  of 
the  air;  s  representing  the  approximate  difference  of  altitude.  It 
is  always  additive 

7.  Table  VII  gives  the  relative  atmospheric  humidity  in  frac 
tions  of  unity.     This  table  is  different  from  any  given  in  Guyot's 
collection ;    for,  though  based  upon  Regnault's  table  of  maximum 
force  of  vapor,  and  so  far  the  same  as  Guyot's,  it  has  been  cal 
culated  with  factors  determined  by  Glaisher. 

8.  Tables  VIII  and  IX  are  intended  to  give  the  correction 

A  log  x  m,  due  to  the  relative  humidity  of  the  stratum  of  air 
between  the  two  stations. 


I48  GEODESY. 


XLV. — Barometrical  Measurement,  &c. — Continued. 

These  hypsometrical  tables  represent  the  full  formula  of  Planta- 
mour.  If,  as  is  often  the  case,  the  observations  for  the  relative 
humidity  are  not  given  with  those  of  the  barometer  and  dry  ther 
mometer,  then  the  tables  III,  (column  B>)  VIII,  and  IX  should 
not  be  used,  but  the  temperature-correction  must  be  calculated 

from  the  formula  A  log  -^  X  ^"t^6-4  with  the  aid  of  column 
H  900 

C  of  Table  III. 

With  the  temperature-term  so  calculated,  the  results  will  differ 
from  those  by  Guyot's  table,  on  account  of  the  different  value 
given  to  the  barometric  constant  of  the  pressure  term. 
5  Abnormal  and  Horary  Oscillations  of  the  Weight  of  the  Atmo 
sphere.— 1\&  first  is  a  gradual  change  generally  extending  over  a 
period  of  two  to  seven  days,  causing  the  barometer  to  rise  or  fall 
gradually  during  that  time,  although  sometimes  more  or  less  sud 
den,  and  occupying  perhaps  but  a  few  hours;  the  second,  a  regu 
lar  horary  oscillation  occurring  at  about  the  same  hours  every  day, 
and  having  a  magnitude  entirely  independent  of  this  gradual 

change. 

The  abnormal  change  usually  extends  over  large  tracts  of  coun 
try,  and  in  settled  weather  the  barometer  rises  and  falls  so  gradu 
ally  that  the  forces  that  produce  the  motion  can  be  separated 
with  more  or  less  accuracy  from  the  horary  changes  by  assuming 
the  portion  of  this  great  wave  corresponding  to  24  hours  to  be  a 
straight  line;  generally  inclined,  however,  since  the  observations 
at  any  time  of  a  barometric  day  differs  from  that  of  the  next  one 
at  the  same  hour. 

To  eliminate  this  movement,  subtract  the  barometric  reading 
(reduced  to  32°)  at  the  beginning  of  one  day  from  that  at.  the 
same  hour  on  the  next  succeeding  one,  and  divide  the  difference 
by  24.     The  result  is  the  correction  for  one  hour,  to  be  applied 
with  its  proper  sign  to  the  hour  succeeding  the  initial  hour.     The 
correction  for  two  hours  is  twice  the  correction  for  one  hour,  etc. 
EXAMPLE.— Barometer  at  7  a.  m.,  August  7,  =    29.  743 
Barometer  at  7  a.  m.,  August  8,  =    29.  487 

Difference  for  24  hours =  +o.  256 

For  I  hour ==+0.0106 

and  correction  at  8  a.  m.,  =  +  o.oi I ;  at  9,  =  +  0.021  ;  at  10,  =  +  0.032,  etc. 
This  correction  Williamson  names  "reduction  to  level." 


BAROMETRICAL  MEASUREMENT  OF    HEIGHTS.  149 


XLV. — Barometrical  Measurement,  &c. — Continued. 

In  the  horary  oscillation  there  are  two  maximum  and  two  min 
imum  points  during  the  24  hours.  Near  the  sea-level  the  barome 
ter  attains  its  maximum  about  9  or  10  a.  m.  In  the  afternoon 
there  is  a  minimum  about  3,  4,  or  5  p.  m.  It  then  rises  until 
from  10  to  midnight,  when  it  falls  again  until  about  4  a.  m.,  and 
again  rises  to  attain  its  morning  maximum,  the  day-fluctuations 
being  the  larger.  The  oscillation  is  greatest  nearer  the  equator 
and  diminishes  toward  the  poles.  Its  amount  within  the  limits 
of  the  United  States  varies  from  40  to  120  thousandths  of  an  inch 
of  the  barometric  column.  It  is  not  equal  at  all  places  of  the 
same  latitude. 

In  a  series  of  barometric  observations  at  any  place,  the  mean 
barometric  reading  is  better  obtained  from  daily  horary  curves, 
by  plotting  the  separate  readings  of  each  day  reduced  to  32°, 
and  corrected  for  the  abnormal  change  by  reduction  to  level. 
These  would  present  an  approximate  horary  curve  for  every  day 
of  the  series,  from  which  erroneous  or  erratic  observations  could 
be  detected  and  rejected  if  necessary. 

EXAMPLE  OF  THE  USE  OF  THESE  TABLES. 

Geneva  and  the  Grand  St.  Bernard. 

-  h  =  28.600  in.       t  =  48°.2  F.      Relative  humidity,  a  —  0.77 
H  =  22.191  in.       t'=  28°.6  F.       Relative  humidity,  a'  =  0.80 


Lat=46°         /  +  f  =  76°.8  F.  0  +  ^=1.57 

Table  II,  with  argument  7i,  gives 27557-3 

Table  II,  with  argument  H,  gives 20903.7 

Difference  =  first  approximate  difference  of  altitude. . .     6653.6 
Table  III,  col.  B,  with  argument  76°.8,  gives  +  0.0130 

0.0130  x  6653. 6  ±=   +86.5 

Second  approximate  difference  of  altitude =  6740.1 

Table  IV,  arguments  46°  lat.  and  6700  feet,  gives —  0.6 

Table  V,  argument  6740,  gives 19.0 

Table  VI,  argument  6700  feet  and  28.6  inches 0.8 


Third  approximate  difference  of  altitude =    6759.3 


J50  GEODESY. 


XLV.  —  Barometrical  Measurement,  &c.  —  Continued. 

Table  VIII,  arguments  22.2  in.  and  28.6  in.,  gives  79 
Table  IX,  arguments  79  and  *j6°.8  ...........  1  1.9 

11.9  x  1.57  —  vapor  correction  =        18.7 

Difference  of  altitude  ............................        6778.0 

The  altitude  by  level  is  stated  to  be  6791.5  feet. 

The  same  readings  of  the  barometer  and  the  same  air-tempera 
ture  being  used,  but  calculating  the1  value  of  the  temperature- 
term  from  column  C,  table  III, 

temperature-term  =  6653.6  x  0.0142  =  94.4 

The  value  of  the  temperature-term,  as  calculated  in  the  above 
example,  increased  by  the  vapor-correction,  is  105.2,  a  larger 
result  than  by  the  method  of  La  Place,  because  the  sum  of  the 
observed  relative  humidities  of  the  stations  was  greater  than  that 
assumed  by  him. 

In  a  dry  climate  the  reverse  would  have  been  the  case. 

Calculation  of  the  same  Observations  by  Guyofs  Tables. 
First  table  of  Guyot  gives,  for  h  .................        27454.4 

First  table  of  Guyot  gives,  for  H  ...........  .  .....        20825.6 

First  approximate   difference  of  altitude  ...........  =    6628.8 


Second  approximate  difference  of  altitude  ..........  =  6733.1 

Table  III  of  Guyot  gives  .......................  —  0.6 

Table  IV  of  Guyot  gives  .......................  19.0 

Table  V  of  Guyot  gives  ........................  0.8 

Difference  of  altitude  ...........................         6752.3 

This  result  is  25.6  less  than  by  the  following  tables,  and  39.2 
less  than  by  the  spirit-level. 

Aneroid  Barometer. 

The  best  aneroids  are,  as  nearly  as  possible,  compensated  by 
the  maker  for  differences  of  temperature,  so  that  the  index  shall 
remain  at  the  same  reading  on  the  dial  when  it  is  heated  and 
cooled,  and  are  intended  to  be  adjusted  to  read  uniformly 
inches  of  mercury  at  a  temperature  of  32°  Fahrenheit  at  the  level 
of  the  sea  in  45°.  latitude.  But  before  using  any  aneroid  for 


BAROMETRICAL  MEASUREMENT  OF    HEIGHTS.  151 


XLV.  —  Barometrical  Measurement,  &c.  —  Continued. 

accurate  observations  it  should  be  tested  under  an  air-pump, 
together  with  a  mercurial  column,  at  a  known  temperature,  and 
its  scale-errors  carefully  noted.  In  many  of  them  there  is  an 
additional  scale  of  altitudes  in  feet  outside  of  the  scale,  corre 
sponding  to  the  inches  of  mercury,  generally  in  the  best  English 
instruments  divided  and  marked  according  to  a  table  prepared 
for  the  purpose  by  Professor  Airy.  There  are  some,  however, 
very  erroneously  marked. 

As  the  aneroid  is  not  affected  in  its  reading  by  the  variation  in 
the  force  of  gravity,  it  needs  no  correction  for  the  latitude,  nor 
for  the  decrease  of  gravity  in  altitude  acting  on  the  mercurial 
column.  The  correction  for  the  decrease  of  gravity  in  altitude 
acting  on  the  density  of  the  air,  and  the  correction  for  humidity, 
remain;  but  the  first  being  small  in  amount,  it  can  be  omitted,  and 
the  second  combined  with  the  correction  for  temperature,  as  is 
done  in  the  formula  of  La  Place. 

The  formula  for  the  aneroid  would  then  be  : 


Z  =  log  jx  60384.3  Eng.  feet 

and  the  tabular  quantities  may  be  taken  from  these  tables. 

Professor  Airy's  table,  made  for  the  purpose  of  graduating 
aneroids  to  a  scale  of  feet,  gives  the  height  of  the  corrected  mer 
curial  column  in  inches  for  each  fifty  feet  of  altitude  at  50° 
Fahrenheit.  The  formula  is  : 

h  t  +  1'  —  looN 

-  ~       -- 


Z  =  log      x  62759  Eng.  feet 


As  it  is  sometimes  convenient  to  have  an  approximate  formula 
that  can  be  used  without  any  tables  whatever,  the  following  may 
be  found  useful  : 

Z=  55°32  H~+7*  Eng<  feet'  at  55°  Fahrenheit> 
with  a  correction  of  i  -  for  each   degree  of   mean   tempera- 

l"O  v) 

ture  above  55°;  or,  nearly  — 

U-/i  .     i 

L  =   CCOOO  TT  —  :  -  ;± 

H  -f  /*     500 

a  formula  easily  remembered,  but  only  useful  for  altitudes  not 
exceeding  3000  feet. 


GEODESY. 


TABLE  I. — Reduction  of  the  English  Baromctet  to  tJie  Freezing- 
Point. 


English  inches. 


0  g 

0  C 

II 

.7-5 

18 

18.5 

19 

19-5 

20 

20.5 

ii' 

o 

+  .045 

+  .046 

+  .047 

+  .049 

+  .O5O 

+  .051 

+  -053 

o 

I 

•043 

•  045 

.046 

.047 

.048 

.049 

.051    1 

I 

2 

.042 

.043 

.044 

•°45 

.046 

.048 

.049 

2 

3 

.040 

.04! 

.042 

.044 

•045 

.046 

.047 

2 

4 

.039 

.040 

.041 

.042 

•043 

.044 

•045 

4 

5 

•037 

.038 

•  039 

.040 

.041 

.042 

•043 

5 

6 

•  035 

.036 

•037 

.038 

•039 

.040 

.041 

6 

7 

.034 

•  .035 

.036 

•°37 

.038 

.039 

.040 

7 

.   8 

.032 

•  033 

•034 

•035 

.036 

•°37 

.038 

8 

9 

.031 

.032 

.032 

•°33 

•034 

•035 

.036 

9 

10 

.029 

.030 

'  .031 

.032 

.032 

•°33 

•034 

10 

ii 

.028 

.028 

.029 

.030 

.031 

.031 

.032 

ii 

12 

.026 

.027 

.027 

.028 

.029 

.030 

.030 

12 

I3 

.024 

.025 

.  026 

.026 

.027 

.028 

.029 

13 

I4 

.023 

.023 

.024 

.025 

.025 

.026 

.027 

!4 

15 

.021 

.022 

.022 

.023 

.024 

.024 

.025 

15 

16 

.020 

.020 

.021 

.O2I 

.022 

.022 

.023 

•16 

I7 

.018 

.019 

.Oig 

.020 

.020 

.021 

.O2I 

17 

18 

.017 

.017 

.017 

.Ol8 

.018 

.019 

.019 

18 

19 

.015 

.015 

.Ol6 

.Ol6 

.017 

.017 

.018 

20 

21 

.013 

.012 

.012 

.OI2 

.013 

.013 

.013 

.014 

21 

23 
24 

25 

.009 
.007 

.006. 

.009 
.007 
.006 

.009 
.007 
.006 

.009 
.008 
.006 

.OIO 

.008 
.006 

.010 
.008 
.006 

.010 

.po8 
.006 

23 
24 

25 

26 

.004 

.004 

.004 

.004 

.004 

.005 

.005 

26 

27 

.002 

.002 

.003 

.003 

.003 

.003 

.003 

27 

28 

+  .OOI 

+  .OOI 

+  .001 

+  .OOI 

+  .001 

+  .OOI 

+  .001 

28 

29 

—  .OOI 

—  .OOI 

—  .001 

—  .OOI 

—  .OOI 

—  .OOI 

—  .OOI 

29 

30 

.002 

.002 

.002 

.003 

.003 

.003 

.003 

3° 

31 

32 
33 

.004 
.005 
.007 

.004 
.006 
.007 

.004 

.006 

.007 

.004 
.006 
.008 

.004 
.006 
.008 

.004 
.006 

.008 

.005 

.006 

.008 

32 
33 

34 

.009 

.009 

.009 

.009 

.OIO 

.OIO 

.OIO 

34 

35 

.010 

.OIO 

.Oil 

.on 

.on 

.012 

.012 

35 

36 

.012 

.012 

.012 

.013 

.013 

.013 

.014 

36 

37 

.013 

.014 

.014 

.014 

.015 

.015 

.Ol6 

37 

38 

.015 

.015 

.016 

.016 

.017 

.017 

.017 

38 

39 

.016 

.017 

.017 

.018 

.018 

.019 

.019 

39 

40 

.018 

.019 

.Oig 

.020 

.020 

.021 

.O2I 

40 

41 

.020 

.020 

.O2I 

.021 

.022 

.022 

.023 

41 

42. 

.021 

.022 

.022 

.023 

.024 

.024 

.025 

42 

43 
44 

.023 
.024 

.023 
.025 

.024 
.026 

.025 
.026 

.025 
.027 

.026 
.028 

.027 
.028 

43 
44 

45 

.026 

.027 

.027 

.028 

.029 

.030 

.030 

45 

46 

.027 

.028 

.029 

.030 

.031 

.031 

.032 

46 

47 

.029 

.030 

.031 

.031 

.032 

•°33 

•034 

47 

48 

.031 

.031 

.032 

•°33 

.034 

•035 

.036 

48 

49 
5° 

.032 
-.034 

•033 
—  •035 

•034 
—  .036 

'    -035 
-•037 

.036 

—  .038 

•037 
—  .038 

.038 
-•°39 

49 
5° 

BAROMETRICAL    MEASUREMENT    OF    HEIGHTS. 


TABLE  I. — Reduction  of  the  English  Barometer  to  the  Freezing- 
Point — Continued. 


Degrees  of 
Fahrenheit. 

English  inches. 

Degrees  of 
Fahrenheit. 

17-5 

18 

I8.5 

J9 

!9-5 

20 

20.5 

51 

-•035 

—  .036 

—  •°37 

-.038 

—  •039 

—  .  040 

-.041 

51 

52 

•037 

.038 

•039 

.040 

.041 

.042 

•°43 

52 

53 

.038 

•°39 

.041 

.042 

•043 

.044 

•045 

53 

54 

.040 

.041 

.042 

•043 

.044 

.046 

.047 

54 

55 

.041 

•043 

.044 

•045 

.046 

.047 

.049 

55 

56 

•  043 

.044 

•045 

.047 

.048 

.049 

.050 

56 

57 

•  045 

.046 

.047 

.048 

.050 

.051 

.052  . 

57 

58 

.046 

.047 

.049 

.050 

.051 

•053 

•054 

58 

59 

.048 

.049 

.050 

.052 

•053 

•055 

.056 

59 

60 

.049 

.051 

.052 

•054 

•055 

.056 

.058 

60 

61 

.051 

.052 

•054 

•°55 

•°57 

.058 

.060 

61 

62 

.052 

•054 

•055 

•057 

.058 

.060 

.O6l 

62 

63 

•  054 

•055 

•057 

•059 

.060 

.062 

.063 

63 

64 

.056 

.057 

•059 

.060 

.062 

.063 

•065 

64 

65 

•  057 

•059 

.060 

.062 

.064 

.065 

.067 

65 

66 

•  059 

.060 

.062 

.064 

.065 

.067 

.069 

•   66 

67 

.060 

.062 

.064 

.065 

.067 

.069 

.071 

67 

68 

.062 

.064 

.065 

.067 

.069 

.071 

.072 

68 

69 

.063 

.065 

.067 

.069 

.071 

.072 

.074 

69 

70 

.065 

.067 

.069 

.070 

.072 

.074 

.076 

70 

71 
72 

.066 
.068 

.068 
.070 

.O/O 
.072 

.072 
.074 

.074 
.070 

.076 
.078 

.078 
.080 

71 

72 

73 

.070 

.072 

'°74 

•075 

.077 

.079 

.O8l 

73 

74 

.071 

•°73 

•075 

.077 

.079 

.081 

.083 

74 

75 

•073 

•075 

.077 

.079 

'  .081 

.083 

.085 

75 

76 

.074 

.076 

.078 

.081 

.083 

.085 

.087 

76 

77 

.076 

.078 

.080 

-.082 

.084 

.087 

.089 

77 

78 

.077 

.080 

.082 

.084 

.086 

.088 

.091 

78 

79 

.079 

.081 

.083 

.086 

.088 

.090 

.092 

79 

80 

.080 

.083 

.085 

.087 

.090 

.092 

.094 

80 

81 

.082 

.084 

.087 

.089 

.091 

.094 

.096 

81 

82 

.084 

.086 

.088 

.091 

•093 

•095 

.098 

82 

83 

.085 

.088 

.090 

.092 

•095 

.097 

.  OO 

83 

84 

.087 

.089 

.092 

.094 

.097 

.099 

.  01 

84 

85 

.088 

.091 

•093 

.096 

.098 

.  IOI 

.  03 

85 

86 

.090 

.092 

•095 

.097 

.  oo 

.103 

•  °5 

86 

87 

.091 

.094 

.096 

.099 

.  02 

.104 

•  °7 

87 

88 

•°93 

•095 

.098 

.  IOI 

•  °3 

.106 

.  09 

88 

89 

.094 

.097 

.  IOO 

.  IO2 

•  05 

.108 

.  ii 

89 

90 

.096 

.099 

.101 

.104 

•  °7 

.  no 

.  12 

9° 

9i 

.097 

.  IOO 

.103 

.I06 

.  09 

.III 

•  14 

91 

92 

.099 

.102 

.105 

.108 

.  10 

•"3 

.  16 

92 

93 

.  IOI 

.103 

.106 

.109 

.  12 

•  "5 

.  18 

•  93 

94 

—  .  IO2 

—  .105 

—  .108 

.III 

•  14 

.117 

.  20 

94 

95 

•"3 

.  16 

.118 

.  21 

95 

96 

.114 

.117 

.  I2O 

.123 

96 

97 

.116 

.119  ' 

.  122 

.125 

97 

98 

.118 

.  121 

.124 

.  127 

98 

99 

-.119 

—  .122 

-.126 

—  .129      99 

154"                                                         GEODESY. 

TABLE  I.  —  Reduction  of  the  English  Barometer  to  the  Freezing- 

Point  —  Continued. 

0-3 

o43 

English  inches. 

|| 

fj 

21 

2I-5 

22 

22.5 

23 

23-5 

24 

II 

o 

+  .054 

+  •055 

+  -055 

4-  .058 

+  .059 

-f  .060 

+  .062 

0 

I 

.052 

•053 

•053 

.056 

.057 

.058 

•059 

I 

2 

.050 

.051 

.051 

•054 

.055 

.056 

•°57 

2 

3    ' 

.048 

.049 

.049 

.052 

•  053 

•°54 

•055 

3 

4 

.046 

.047 

.047 

.050 

.051 

.052 

•053 

4 

5 

.044 

•045 

•°45 

.048 

.049 

.050 

.051 

5 

6 

.042 

.044 

.044 

.046 

.047 

.048 

.049 

6 

7 

.041 

.042 

.042 

.044 

.044 

•045 

.046 

7 

8 

•039 

.040 

.040 

.041 

.042 

•043 

.044 

8 

9 

•°37 

.038 

.038 

•°39 

.040 

.041 

.042 

9 

10 

•  035 

»    .036 

•036 

•°37 

.038 

•°39 

.040 

10 

ii 

•033 

•034 

•034 

.035     |          .036 

•°37 

.038 

ii 

12 

.031 

.032 

.032 

•033              -°34 

•035 

.036 

12 

13 

.029 

.030 

.030 

.031               .032 

•«33 

•033 

13 

T4 

.027 

.028 

.028 

.029     j          .030 

.031 

.031 

15 

.025 

.026 

.026 

.027     1          .028 

.029 

.029 

15 

16 

.024 

.024 

.024 

.025     |          .026 

.026 

.027 

16 

17 

.022 

.022 

.022 

.023 

.024 

.024 

.025 

iy 

18 

.020 

.020 

.020 

.021 

.022 

.022 

.023 

18 

I9 

.018 

.018 

.018 

.019 

.020 

.020 

.020 

ig 

20 

.016 

.016 

.016 

.017 

.018 

.018 

.018 

20 

21 

.014 

.014 

.014 

.015 

.Ol6 

.Ol6 

.016 

21 

23 

.010 

.013 

.Oil 

.013 

.Oil 

.013 

.Oil 

.Oil 

.012 

.012 

23 

24 

.008 

.009 

.009 

.009 

.009 

.OIO 

.010 

24 

25 

.007 

.007 

.007 

.007 

.007 

.007 

-.008 

25 

26 

.005 

.005 

.005 

.005 

.005 

.005 

.005 

26 

27 

.003 

.003 

.003 

.003 

.003 

.003 

.003 

27- 

28 

+  .001 

+  .001 

+  .001 

+  .001 

+  .001 

4-  .001 

+  .001 

28 

29 

—  .001 

—  .001 

—  .001 

—  .001 

—  .001 

—  .001 

—  .001 

29 

3° 

.003 

.003 

.003 

.003 

.003 

.003 

.003 

30 

31 

.005 

.005 

.005 

.005 

.005 

.005 

.005 

3r 

32 

.007 

.007 

.007 

.007 

.007 

.007 

.008 

32 

33 

.008 

.009 

.009 

.009  . 

.009 

.009 

.010 

33 

34 

.010 

.on 

.on 

.on 

.Oil 

.012 

.012 

34 

35 

.012 

.013 

.013 

.013 

.013 

.014 

.014 

35 

05 

.01  <5 

,Ol6 

,Ol6 

tf 

3° 
37 

.016 

.016 

.017 

.017 

.018 

.Ol8 

.018 

J^ 

37 

38 

.018 

.018 

.019 

.019 

.020 

.020 

.020 

38 

39 

.020 

.020 

.021 

.021 

.022 

.022 

.023 

39 

40 

.022 

.022 

.023 

.023 

.024 

.024 

.025 

40 

4i 

.023 

.024 

.025 

.025 

.026 

.026 

.027 

41 

42 

.025 

.026 

.027 

.027 

.028 

.028 

.029 

42 

43 

.027 

.028 

.029 

.029 

.030 

.031 

.031 

43 

44 

.029 

.030 

.031 

.031 

.032 

•°33 

•°33 

44 

45 

.031 

.032 

.032 

•  033 

•034 

•035 

•°35 

45 

46 

•033 

•034 

•  034 

•  035 

.036 

•037 

.038 

46 

47 

•035 

.036 

.036 

•  037 

.038 

•039 

.040 

47 

48 

•037 

.038 

.038 

•039 

.040 

.041 

.042 

48 

49 

•°39 

•°39 

.040 

.041 

.042 

•043 

.044 

49 

50 

—  .040 

—  .041 

—  .042 

-.043 

—  .044 

—  •045 

—  .046 

50 

BAROMETRICAL    MEASUREMENT    OF    HEIGHTS. 


TABLE   I. — Reduction  of  the  English  Barometer  to  the  Freezing- 
Point — Continued. 


-~  .*-» 

0-5 

CBrt 

sg 

Q  b 

English  inches. 

°'S 

B-a 

£?S 

fra 

p£ 

21 

21.5 

22 

22.5 

23 

23-5 

24 

51 

-.042 

--043 

-.044 

-•045 

—  .046 

—  .047 

-.048 

sr 

52 

.044 

•045 

.046 

.047 

.048 

.049 

.050 

52 

53 

.046 

.047 

.048 

.049 

.050 

.051 

•053 

53 

54 

.048 

.049 

.050 

.051 

.052 

•054 

•055 

54 

55 

.050 

.051 

.052 

•053 

•055 

.056 

•°57 

55 

56 

.052 

••053 

•054 

•055 

•057 

.058 

•°59 

56 

57 

•  °54 

•°55 

.056 

•057 

^059 

.060 

.061 

57 

58 

•  055 

•057 

.058 

•059 

.O6l 

.062 

.063 

58 

59 

•°57 

•059 

.060 

.061 

.063 

.064 

.065 

59 

60 

•059 

.O6l 

.062 

•063  • 

.065 

.066 

.068 

60 

61 

.061 

.062 

.064 

.065 

.067 

.068   |      .O7O 

61 

62 

.063 

.064 

.066 

.067 

.069 

.070 

.072 

62 

63 

.065 

.066 

.068 

.069 

.071 

.072 

.074 

63 

64 

.067 

.068 

.070 

.071 

•°73 

•075 

.076 

64 

65 

.068 

.070 

.072 

•°73 

•°75 

.077 

.078 

65 

66 

.070 

.072 

.074 

•°75 

.077 

.079 

.080 

66 

67 

.072 

.074 

.076 

.077 

.079 

.O8l 

.083 

67 

68 

.074 

.076 

.078 

.079 

.O8l 

.083 

.085 

68 

69 

.076 

.078 

.080 

.081 

.083 

•085 

.067 

69 

70 

.078 

.030 

.082 

.083 

.085 

.087 

.089 

70 

7i 

.080 

.082 

.083 

.085 

.087         .089 

.091 

71 

72 

.082 

.084 

.085 

.087 

.089         .Ogi 

•°93 

72 

73 

•083 

.085 

.087 

.089 

.091         .093 

•095 

73 

74 

.085 

.087 

.089 

.091 

.093         .095 

.097 

74 

75 

.087 

.089 

.091 

•°93 

.095   j      .098 

.100 

75 

76 

.089 

.09I 

•093 

•095 

.098  1    .100 

.  IO2 

76 

77 

.091 

•°93 

•095 

.097 

.100         .102         .104 

77 

73 

•°93 

•095 

.097 

.099 

.  102      .  104      .  106 

78 

79 

•095 

.097 

.099 

.101 

.104 

.  106      .  108 

79 

80 

.096 

.099 

.  IOI 

.103 

.106 

.108      .no 

80 

81 

.098 

.101 

.103 

.105 

.108      .no 

.  112 

81 

82 

.  IOO 

.103 

.105 

.107 

.110        .112 

•"S 

82 

83 

.  IO2 

.105 

.107 

.109 

.  112 

.114 

.117 

83 

84 

.104 

.106 

.109 

.in 

.114 

.116 

.119 

84 

85 

.106 

.108 

.III 

.113      .116 

.118 

.  121 

85 

86 

.108 

.  no 

•Ir3 

.115      .118  j    .120 

.123 

86 

87 

.  no 

.  112 

•115 

.117 

.120  i    .123  I    .125 

87 

88 

.  Ill 

.114 

.117 

.  119 

.  122 

.125 

.127 

88 

89 

•"3 

.116 

.119 

.121         .124 

.127 

.129 

89 

90 

•ii5 

.Il8 

.121 

.123 

.126 

.129 

.132 

90 

Qi 

.117 

.  I2O 

.123 

•125 

.128         .131 

•134 

91 

92 

.119 

.  122 

.124 

.I27 

•13°        -133 

.136 

92 

93 

.  121 

.124 

.126 

.129 

•132   j      .135 

.138 

93 

94 

.123 

.125 

.128 

•131 

•134        .137 

.140 

94 

95 

.124 

.127 

.130 

•133 

•136   |      .139 

.142 

95 

96 

.126 

.129 

.132 

•135 

.138   |      ,I4I 

.144 

96 

97 

.128 

•J31 

.134 

•137 

.140 

•*43 

.146 

97 

98 

.I30 

•133 

.136 

•139 

.142 

.145      .149 

98 

99 

-.132 

—  .I3S 

-.138 

—  .141 

-.144 

-.148 

—  -iS1 

99 

'56 


GEODESY. 


TABLE  I. — Reduction  of  the  English  Barometer  to  the  Freezing- 
Point — Continued. 


ll 

English  inches. 

0-5 

»! 

Oj  Jn 

Qr^ 

240 

25 

25-5 

26 

26.5 

27 

27-5 

|| 

0 

+  .063 

+  .064 

+  -065 

+  .067 

+  .068 

+  .069 

+  .071 

o 

I 

.061 

.062 

.063 

.064 

.066 

.067 

.068 

I 

2 

.058 

.060 

.061 

.062 

.063 

.064 

.066 

2 

3 

.056 

•°57 

.058 

.060 

.061 

.062 

.063 

3 

4 

•  054 

•055 

.056 

•057 

.058 

.060 

.061 

4 

5 

.052 

•053 

•054 

•°55 

.056 

•057 

.058 

5 

6 

.050 

.051 

.052 

•053 

•054 

•°55 

.056 

6 

7 

.047 

.048 

.049 

.050 

.051 

.052 

•053 

7 

8 

•045 

.046 

.047 

.048 

.049 

.050 

.051 

8 

9 

•  043 

.044 

•045 

.046 

.046 

.047 

.048 

9 

10 

.041 

.042 

.042 

•043 

.044 

•°45 

.046 

10 

ii 

•039 

•039 

.040 

.041 

.042 

.042 

•043 

ii 

12 

.036 

•°37 

.038 

•°39 

•°39 

.040 

.041 

12 

I3 

•034 

•035 

.036 

.036 

•037 

.038 

.038 

I  s 

J4 

.032 

•°33 

•033 

•034 

•035 

•035 

.036 

14 

15 

.030 

.030 

.031 

.032 

.032 

•°33 

•033 

15 

16 

.028 

.028 

.029 

.029 

.030 

.030 

.031 

16 

18 

.025 

.026 

.026 

.027 

.027 

.028 

.028 

18 

.021 

.021 

.022 

.022 

.023 

.023 

.023 

19 

20 

.019 

.019 

.Dig 

.O2O 

.020 

.021 

.021 

20 

21 

.017 

.017 

.017 

.018 

.018 

.018 

.019 

21 

22 

.014 

.015 

.015 

.015 

.015 

.016 

.016 

22 

23 

.012 

.012 

.013 

.013 

.013 

.013 

.014 

23 

25 

.008 

.008 

.008 

.008 

.008 

.009 

.009 

24 
25 

26 

.006 

.006 

.006 

.006 

.006 

.006 

.006 

26 

27 

.003 

.003 

.003 

.004 

.004 

.004 

.004 

27 

28 

+  .001 

+  .001 

+  .001 

+  .OOI 

4-  .001 

+  .001 

+  .001 

28 

29 

—  .001 

—  .001 

—  .001 

—  .001 

—  .001 

-.001 

—  .001 

29 

30 

.003 

.003 

.003 

.003 

.004 

.004 

.004 

30 

31 

.005 

.006 

.006 

.006 

.006 

.006 

.006 

31 

32 

.008 

.008 

.co8 

.008 

.008 

.008 

.009 

32 

33 

.010 

.010 

.010 

.010 

.Oil 

.Oil 

.Oil 

33 

34 

.012 

.012 

.013 

.013 

.013 

•  -OI3 

.014 

34 

35 

.014 

.015 

.015 

.015 

.015 

.016 

.016  . 

35 

36 

.016 

.017 

.017 

.017 

.018 

.018 

.018 

36 

37 

.Dig 

.Dig 

.019 

.020 

.020 

.021 

.021 

37 

38 

.O2I 

.021 

.022 

.022 

.023 

.023 

.023 

38 

39 

•023 

.024 

.024 

.024 

.025 

.025 

.026 

39 

40 

.025 

.026 

.©26 

.027 

.027 

.028 

.028 

40 

41 

.027 

.028 

.029 

.029 

.030 

.030 

.031 

4i 

42 

.030 

.030 

.031 

.031 

.032 

•033 

•°33 

42 

43 

.032 

.032 

•033 

•  034 

•034 

•035 

.036 

43 

44 

•034 

•035 

•035 

.036 

•037 

•037 

.038 

44 

45 

.036 

•037 

.038 

.038 

•039 

.040 

.041 

45 

46 

.038 

•039 

.040 

.041 

.042      .042 

•043 

46 

47 

.041 

.041 

.042 

•  043 

.044 

•°45 

.046 

47 

'  48 

•043 

.044 

.044 

•  045 

.046 

.047 

.048 

48 

49 

•045 

.046 

.047 

.048 

.049 

.050 

49 

50 

-.047 

—  .048 

—  .049 

—  .050 

-.051 

—  .052 

-•053 

50 

BAROMETRICAL    MEASUREMENT    OF    HEIGHTS.  157 

TABLE  I. — Reduction  of  the  EnglisJi  Barometer  to  the  Freezing- 
Point — Continued. 


0-55 

gc 

English  inches. 

I 

U_rt 

(31 

24-5 

25 

25-S' 

26 

26.5 

27 

27-5 

II 

51 

—  .049 

—  .050 

-.051 

—  .052  • 

—  •053 

—  •054 

—  •°55 

51 

52 

.052 

•053 

•054 

•°55 

.056 

•°57 

.058 

52 

53 

•  054 

•055 

.056 

•057 

.058 

•°59 

.060 

53 

54 

55 

.056 
.058 

•057 
•059 

.058 
.060 

•°59 
.062 

.060 
.063 

.062 

.064 

.063 
.065 

54 
55 

56  ' 

.060 

.061 

.063 

.064 

.065 

.066 

.068 

56 

57 

.062 

.064 

.065 

.066 

.068 

.069 

.070 

58 

.065 

.066 

.067 

.069 

.070 

.071 

•°73 

58 

59 
60 

.067 
.069 

.068 
.070 

.070 
.072 

.071 
•073 

.072 
•°75 

.074 
.076 

•°75 
.077 

59 
60 

61 

.071 

•°73 

.074 

.076 

.077 

.078 

.080 

61 

62 

•  073  . 

•075 

.076 

.078 

.079 

.081 

.082 

62 

63 
64 

.65 

.076 
.078 
.080 

.077 
.079 
.082 

.079 
.081 
.083 

.080 
.082 
.085 

.082 
.084 
.086 

.083 
.086 
.088 

.085 
.087 
.090 

s 

65 

66 
67 

.082 
.084 

.084 
.086 

.085 
.088 

.087 
.089 

.089 
.091 

.090 
•093 

.092 
•°95 

66 
67 

68 

.086 

.088 

.090 

.092 

•093 

•°95 

.097 

68 

69 

.089 

.090 

.092 

.094 

.096 

.098 

.099 

6q 

70 

.091 

•°93 

•095 

.096 

.098 

.100 

.  102          70 

71 

•  093 

•095 

.097 

.099 

.  IOI 

.102 

.104         7I 

72 

•  095 

.097 

.099 

.  IOI« 

.103 

.105 

.107          72 

73 

-.097 

.099 

.  IOI 

.103 

.105 

.107 

•  I09  i    73 

74 

.  IOO 

.  IO2 

.104 

.106 

.108 

.110 

.112   i       7J. 

75 

.  IO2 

.104 

.106 

.108 

.110 

.  112 

MI4 

75 

76 

.104 

.106 

.108 

.110 

.  112 

.114 

.117          76 

77 

.106 

.108 

'.  IIO 

•IJ3 

•"5 

.117 

.119 

77 

78 

.I08 

.  no 

•"3 

•  "5 

.117 

.119 

.  121 

78 

79 

.  no 

•"3 

•US 

.117 

.119 

.  122 

.124  '     79 

80 

•  113 

•US 

.117 

.119 

.122 

.124 

.126 

So 

81 

•  us 

.117 

.119 

.122 

.124 

.126 

.129 

81 

82 

.117 

.119 

.  122 

.124 

.126 

.129 

82 

83 
84 

85 

.119 

.121 

.123 

.  122 
.124 
.126 

.124 
.126 
.128 

.126 
.129 

•  I31 

.129 
•134 

•134 

•134 
.136 

83 
84 
85 

86 
87 

.126 
.128 

.128 
.130 

•I31 
•!33 

•  133 
.136 

.136 
.138 

.138 
.141 

.141 

.  143 

86 
87 

88 

.130 

•133 

•135 

.138 

.141 

•   -143 

146 

88 

89 

.132 

•135 

.138 

.140 

•143 

.146 

.148 

89 

90 

•I34 

•137 

.140 

•143 

•145 

.148 

90 

9i 

.136 

•139 

.142 

•145 

.I48 

.150 

•153 

91 

92 

•!39 

.141 

.144 

.147 

.150 

•153 

.156 

92 

93 

.141 

.144 

.147 

.149 

.152 

'155 

.158 

93 

94 

•143 

.146 

.149 

.152 

•155 

.158 

.160 

94 

95 

•  145 

.148 

•151 

•  154 

•I57 

.160 

.163 

95 

96 

.147 

.150 

•153 

.156 

•  159 

.162 

.165 

96 

97 

.149 

•!53 

.156 

•  159 

.162 

.165 

.168 

97 

98 

.152 

•155 

•158 

.161 

.164 

.167 

.  170 

98 

99 

I 

—  •154 

—  •I57 

-.160 

-.163 

-.166 

—  .170 

99 

GEODESY. 


TABLE  I. — Reduction  of  the  English  Barometer  to  the  freezing- 
Point — Continued. 


o| 

English  inches. 

<**  -J 

Si  £ 

&l£ 

ol 

28 

28.5 

29 

29-5 

30 

3°-5 

31 

Q' 

o 

+  .072 

+  -°73 

+  .074 

+  .076 

+  .077 

+  .078 

+  .080 

o 

I 

.069 

.071 

.072 

•°73 

.074 

•075 

.077 

I 

2 

.068 

.069 

.070 

.072 

•073 

.074 

2 

3 
4 
5 

!o62 

•059 

.065 
.063 
.060 

.067 
.064 
.061 

.068 

.065 
.062 

.069 

.066 

.063 

.070 

.067 
.064 

.071 
.068 
.066 

3 
4 
5 

6 
7 

•057 
•  054 

.058 
•055 

•  059 
.056 

.060 

•057 

.061 
.058 

.062 

.063 

.060 

6 
7 

8 

.052 

•°53 

.053 

•  054 

•055 

.056 

•  057 

8 

9 

.049 

.050 

.051 

.052 

.053 

•053 

•054 

9 

10 

.047 

.047 

.048 

.049 

.050 

.051 

.052 

10 

ii 

.044 

•045 

.046 

.046 

.047 

.048 

.049 

ii 

12 

.042 

.042 

•043 

.044 

.044 

•045 

.046 

12 

J3 

•  039 

.040 

.040 

.041 

.042 

.042 

.043 

13 

15 

.036 

.034 

•°37 
•035 

.038 
•  035 

.038 
.036 

•  039 
.036 

.040 
•037 

.040 
.038 

15 

16 

.031 

.032 

•033 

•°33 

•  034 

.034 

.035 

16 

\l 

.029 
.026 

.029 
.027 

.030 
.027 

.030 
.028 

.031 
.028 

.032 
.029 

.032 
.029 

i? 
18 

29 

.024 

.024 

.025 

.025 

.026 

.026 

.027 

19 

20 

.021 

.022 

.022 

.023 

.023 

.023 

.024 

20 

21 

.019 

.019 

.020 

.020 

.020 

.021 

.021 

21 

22 

.Ol6 

.017 

.017 

.017 

.018 

.018 

.018 

22 

23 

.014 

.014 

.OI4 

.015 

.015 

.015 

.015 

23 
24 

24 

25 

.009 

.009 

.OOg 

.009 

.009 

.010 

.010 

25 

26 

.006 

.006 

.007 

.007 

.007 

.007 

.007 

26 

27 

.004 

,004 

.004 

.004 

.004 

.004 

.004 

27 

28 

+  .OOI 

+  .OOI 

+  .OOI 

+  .001 

+  .001 

+  .OOI 

+  .OOI 

28 

29 

—  .001 

—  .OOI 

—  .OOI 

—  .001 

—  .OOI 

—  .OOI 

—  .OOI 

29 

30 

.004 

.004 

.004 

.004 

.004 

.004 

.004 

30 

.006 

.006 

.006 

.007 

.007 

.007 

.007 

31 

32 

.009 

.009 

.009 

.009 

.009 

.010 

.010 

32 

33 
34 

.Oil 

.014 

fOI4 

.014 

.015 

.015 

.015 

.015 

34 

35 

.016 

.017 

.017 

.017 

.017 

.018 

.018 

35 

36 

.019 

.019 

.019 

.020 

.020 

.020 

.021 

36 

.021 

.022 

.022 

.022 

.023 

.023 

.024 

37 

38 

.024 

.024 

.025 

.025 

.026 

.026 

.026 

38 

39 

.026 

.027 

.027 

.028 

.028 

.029 

.029 

39 

40 

.029 

.029 

.030 

.030 

.031 

.031 

.032 

40 

41 
42 

43 

.031 

.034 
.036 

.032 
.034 
•037 

.032 
•035 
.038 

•  033 

.036 

.038 

.034 

.036 

•039 

•034 
.037 

.040 

.035 
•037 

.040 

41 

42 
43 

44 

•°39 

.040 

.040 

.041 

.042 

.042 

•  043 

44 

45 

.041 

.042 

•043 

.044 

.044 

•045 

.046 

45 

46 

.044 

•045 

•045 

.046 

.047 

.048 

.049 

46 

47 

.046 

.047 

.048 

.049 

.050 

.050 

.051 

47 
.c 

48 
49 
5° 

.049 
.051 
-•054 

.050 

.052 
-.055 

.051 
•°53 
—  .056 

.051 
.054 
-.057 

.052 
•  055 

-.058 

.053 
.056 

—  •059 

.054 
•057 
-.060 

40 
45 
5C 

BAROMETRICAL   MEASUREMENT   OF    HEIGHTS. 


TABLE  I. — Reduction  of  the  English  Barometer  to  the  Freezing- 
Point — Continued. 


if 

English  inches. 

11 

tjs 

II 

^ 

28 

28.5 

29 

29-5 

30 

30-5 

3i 

51 

-.056 

-.057    —.058 

—  •059 

-.060 

-.061 

-.062 

'  51 

52 

•059 

.060 

.061 

.062 

.063 

.064 

.065 

52 

53 

.061 

.  062      .  064 

.065 

.066 

.067 

.068 

53 

54 

.064 

.065  i    .066 

.067 

.068 

.070 

.071 

54 

55 

.066 

.068      .069 

.670 

.071 

.072 

•073 

55 

56 

.069 

.070      .071 

•073 

.074 

•075 

.076 

56 

57 

.071 

.073      .074 

•075 

.076 

.078 

.079 

57 

58 

.074 

.075      .076 

•078 

.079 

.080 

.082 

58 

59 

.076 

.078      .079 

"080 

.082 

•  083 

.085 

59 

60 

.079 

.080      .082 

'.083 

.084 

.086 

.087 

60 

61 

.081 

083      .084      .086 

.087 

089 

.090 

61 

62 

.084 

.085      .087 

.088 

.090 

.091 

•°93 

62 

63 

.086     .088     .089 

.091 

.092 

.094 

.096 

63 

64 

io8g      .090      .092 

.094 

•095 

.097 

.098 

64 

f>5 

.091      .093      .095 

.096 

.098 

.099 

,  IOI 

6s 

66 

.094      .095  |    .097 

.099 

.101 

.  IO2 

.104 

66 

67 

.096      .098 

.100 

.  IOI 

.103 

.105 

.107 

67 

68 

.099      .101 

.  IO2 

.104 

.106 

.108 

.  109 

68 

69 

.101      .103    '  .105 

.107 

.109 

.  no 

.112 

69 

7° 

.  104  ;    .  106      .  107  i    .  109 

.  Ill 

•"3 

•"5 

70 

7i 

.106  ;    .108      .no      .112 

.114 

.116 

.118 

71 

72 

.109      .in      .113 

.115      .117 

.118 

.120 

72 

73 
74 

.III   j      .!I3 
.114        .Il6 

•  MS 

.118 

.117 

.  I2O 

.119 

.  122 

.  121 

.124 

.123 
.120 

73 
74 

75 

.Il6        .Il8         .I2O 

.  122 

.125 

.127 

.129 

75 

76 

.119   i     .121        .123 

.125 

.127 

.129 

•131 

76 

77 

.121   !      .123 

.  126 

.128 

.130 

.132 

•134 

77 

78 

.124   ;     .126       .128 

.130 

•133 

•I3S 

•137 

78 

79 

.126 

.128 

•  *3* 

•133 

•135 

•  137 

.140 

79 

80 

.129    ;       .131 

•*33 

.136 

.138 

.140 

•143 

80 

81 

•I31      -133      -136 

.I38 

.141 

.143 

•145 

81 

82 

•  134 

.136        .138 

.141 

•I43 

.146 

.148 

82 

83 

.136 

•139 

.141 

•143 

.146 

.148 

•IS1 

83 

84 

.139    .141 

.144 

.146 

.148 

84 

85 

.141   ;     .144 

.146 

.149 

•15*4 

'156 

85 

86 

.144   '     .146       .149 

•151 

•154 

.156 

-X59 

86 

87 

.146 

.149 

•151 

•154 

.156 

.  162 

87 

88 

.149 

.151 

•154 

.156 

•159 

.162 

.165 

88 

89 

•  I5I 

.154        .156 

•159 

.162 

.164 

.167 

89 

90 

•153      -156 

•159 

.162 

.164 

.167 

.170 

90 

91 

.156      .159      .162 

.164 

.167 

.170 

•I73 

91 

92 
93 
94 

,158 
.161 
•163 

'.\6\ 
.166 

.164 
.167 
.169 

.167 
,I70 
-I72 

.170 
.172 
•J75 

•I73 
!i78 

.175 

92 
93 
94 

95 

.166;    .  169 

.172 

•175 

.178 

.181 

'183 

95 

96 

.168      .171 

.174 

.177 

.180 

.183 

.186 

96 

97 

.171      .174 

.177 

.ISO 

.183 

.186 

.189 

97 

98 
99 

•173 
-.176 

.176 
-.179 

.180 
-.182 

.I83 
-.185 

.186 
-.188 

.189 
-.191 

.191 
-.194 

98 
-99 

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GEODESY. 


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BAROMETRICAL    MEASUREMENT    OF    HEIGHTS.  l6l 


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BAROMETRICAL    MEASUREMENT    OF    HEIGHTS.  163 


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BAROMETRICAL    MEASUREMENT    OF    HEIGHTS.  165 


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GEODESY. 


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ONCO    —vo    -3-  ro  01   M    O     1                           ON  t^vo   -"i-  ro  04   O    O>  — 
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BAROMETRICLL  MEASUREMENT  OF    HEIGHTS. 


I67 


TABLE  III.  —  Correction  for  Temperature. 
DI£  =  A  log  ~ X  B  or  C. 


B 

C 

B 

C 

t  +  t' 

*  +  t1  —  64 

t  +  1'  -  64 

t  +  f 

t  +  t'  —  64 

t  +  t'  —  64 

982.2647 

900 

982.2647 

900 

3° 

-0.0346 

—0.0378 

70 

+  o  .  006  1 

+  0.0067 

3i 

•  0336 

.0367 

7i 

.0071 

.0078 

32 

.0326 

•0356 

72 

.0081 

.0089 

33 

.0316 

•  0344 

73 

.0092 

.0100 

34  ' 

•  0305 

•  0333 

74 

.0102 

.OIII 

35 

.0295 

.0322 

75 

.0112 

.OI22 

36 

.0285 

.0311 

76 

.OI22 

•0133 

37 

.0275 

.0300 

77 

.0132 

.0144 

38 

.0265 

.0289 

78 

.0142 

.0156 

39 

•  0255 

.0278 

79 

•0153 

.0167 

40 

.0244 

.0267 

80 

.0163 

.0178 

41 

.0234 

.0256 

81 

.0173 

.0189 

42 

.0224 

.0244 

82 

.0183 

.0200 

43 

.0214 

.0233 

83 

.0193 

.02  1  1 

44 

.0204 

.0222 

84 

.O2O4 

.0222 

45 

.0193 

.0211 

85 

.0214 

.0233 

46 

.0183 

.0200 

86 

.0224 

.0244 

47 

•0173 

.0189 

87 

.0234 

.0256 

48 

.0163 

.0178 

88 

.0244 

.0267 

49 

•OI53 

.0167  . 

89 

•0255 

.0278 

So 

.0143 

.0156 

90 

.0265 

.0289 

5i 

.0132 

.0144 

91 

.0275 

.0300 

52 

.0122 

•0133 

92 

.0285 

.0311 

53 

.0112 

.0122 

93 

.0295 

.0322 

54 

.0102 

.OIII 

94 

•0305 

•0333 

55 

.0092 

.0100 

95 

.0316 

•0344 

56 

.0081 

.0089 

96 

.0326 

•0356 

57 

.0071 

.0078 

97 

.0336 

.0367 

58 

.0061 

.0067 

98 

.0346 

.0378 

59 

.0051 

.0056 

99 

•°356 

.0389 

60 

.0041 

.0044 

IOO 

.0366 

.0400 

61 

.0030 

.0033 

101 

•0377 

.0411 

62 

.0020 

.0022 

1  02 

.0387 

.0422 

63 

—  o.ooio 

—  O.OOII 

103 

•0397 

•0433 

64 

.0000 

.0000 

104 

.0407 

.0444 

65 

+  O.OOIO 

+  O.OOII 

I05 

.0417 

.0456 

66 

.0020 

.0022 

106 

.0428 

.0467 

67 

.  .0030 

.0033 

107 

.0438 

.0478 

63 

.0041 

.0044 

108 

.0448 

.0489 

69 

+  0.0051 

+  0.0056 

109 

+  0.0458 

+  O.O5OO 

i68 


GEODESY. 


TABLE  III. — Correction  for  Temperature — Continued. 
Dn  —  A  log  ~  x  B  or  C. 


15 

C 

B 

C 

t  +  t' 

t  +  t1  -  64 

/  +  t'  —  64 

t  +  t' 

t  +  t'-  64 

t  +  t'  —  64 

982.2647 

900 

982.2647 

900 

no 

+  0.0468 

+  0.0511 

150 

+  0.0875 

+  0.0958 

III 

.0478 

.0522 

151 

.0886 

.0967 

112 

.0489 

.0533 

152 

.0896 

.0978 

113 

.0499 

•°544 

153 

.0906 

.0989 

114 

.0509 

•0556 

J54 

.0916 

.1000 

II5 

.0519 

.0567 

155 

.0926 

.ion 

116 

.0529 

.0578 

156 

•°937 

.1022 

117 

.0540 

.0589          157 

.0947 

•1033 

118 

•0550 

.0600 

158 

•°957 

.1044 

119 

.0560 

.0611 

159 

.9967 

.1056 

I2O 

.0570 

.0622 

1  60 

.0977 

.  1067 

121 

.0580 

.0633 

161 

.0987 

.1078 

122 

.0590 

.0644          162 

.0998 

.1089 

I23 

.0601 

.0656 

163 

.1008 

.  IIOO 

124 

.0611 

.0667 

164 

.1018 

.1111 

125 

.0621 

.0678 

165 

.1028 

.  1122 

126 

.0631 

.0689 

1  66 

.1038 

•«33 

127 

.0641 

.0700 

167 

.1049 

.1144 

128 

.0651 

.0711 

168 

.1059 

•  1156 

I29 

.0662 

.0722 

169 

.1069 

.1167 

I30 

.0672 

•0733 

170 

.1079 

.1178 

I3I 

.0682 

.•0744 

171 

.1089 

.1189 

132 

.0692 

.0756 

172 

.1099 

.1200 

*33 

.0702 

.0767 

173 

.  1109 

.1211 

134 

.0713 

.0778 

174 

.  II2O 

.1222 

135 

.0723 

.0789 

175 

.1130 

•1233 

136 

•°733 

.0800 

176 

.  1140 

.1244 

137 

•°743 

.0811 

177 

.1150 

.1256 

138 

•c/53 

.0822 

178 

.  1160 

.  1267 

139 

.0763 

•o333 

179 

.1171 

.1278 

140 

.0774 

.0844 

1  80 

•  .1181 

.1289 

141 

.0784 

.0856 

181 

.1191 

.  I3OO 

142 

.0794 

.0867 

182 

.  I2OI 

.1311 

143 

.0804 

.0878 

183 

.1211 

.1322 

144 

.0814 

.0889 

184 

.  1222 

•1333 

J45 

.0825 

.0900 

185 

.1232 

.1344' 

146 

•0835 

.0911 

186 

.1242 

•fjS6 

M7 

.0845 

.0922 

187 

.1252 

•1367 

148 

•0855 

•°933 

188 

.1262 

•1378 

149 

+  0.0865 

+  0.0944 

189 

+  o.  1272 

+  0.1389 

BAROMETRICAL  MEASUREMENT  OF  HEIGHTS.  169 


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170 


GEODESY. 


TABLE  -V. — Correction  for  Decrease  of  Gravity  on  a  Vertical  actin- 
on  the  Density  of  the  Mercurial  Column. 


DIV  =  60384.3  log  4  4. 


60384.3  log      -  +  52252 


20886860 


Approx  . 
cliff,  of  alt. 

OOO 

100 

200 

300 

400 

500 

600 

700 

800 

900 

Feet. 

!  Feet. 

Feet. 

Feet. 

Feet. 

feet. 

Feet. 

Feet. 

Feet. 

Feet. 

IOOO 

2-5 

2.8 

3-i 

3-3 

3-6 

3-9 

4.1 

4-4 

4-7 

4-9 

2OOO 

5-2 

5-5 

5-7 

6.0 

6-3 

6.6 

6.8 

7  •  * 

7-4 

7-7 

3OOO 

7-9 

8.2 

8-5 

8.8 

9.1 

9-3 

9.6 

9-9 

IO.2 

10.5 

4OOO 

10.8 

II.  I 

11.4 

ii.  6 

11.9 

12.2 

12.5 

12.8 

I3.I 

J3-4 

5000 

13-7 

14.0 

14.3 

14.6 

14.9 

J5-2 

15-5 

15-8 

16.1 

16.4 

6OOO 

16.7 

17.0 

i7-4 

17.7 

18.0 

I8.3 

18.6 

18.9 

19.2 

19-5 

JOOO 
8000 

19.9 

20.2 

20.5 

20.8 

21  .  I  . 

21.5 

21.8 

22.  I 

22.4 

22.8 

oA   T 

9OOO 

26.4 

26.7 

27.  i 

20.  1 

10000 

29.8 

30.2 

30-5 

30.8 

31-2 

31-5 

31-9 

32.2 

29.  i 
32.6 

29.5 

33-o 

I  IOOO 

33-3 

33-7 

34-o 

34-4 

34-7 

35-1 

35-5 

35-8 

36.2 

36.5 

I2OOO 

36.9 

37-3 

37-6 

38.0 

38.4 

38.8 

39-i 

39-5 

39-9 

40.2 

13000 

40.6 

41.0 

41.4 

41.7 

42,  i 

42.5 

42.8 

43-3 

43-6 

44.0 

14000 

44-4 

44-8 

45-2 

45-6 

46.0 

46.3 

46/7 

47.1 

47-5 

47-9 

15000 

43.3 

48.7 

49.1 

49-5 

49.9 

50.3 

50.7 

51-1 

5*  -5 

51-9 

16000 

52.3 

52.7 

53-i 

53-5 

53-9 

54-3 

54-7 

55-i 

55-5 

56.0 

17000 

56.4 

56.8 

57-2 

57-6 

58.0 

58.4 

58.9 

59-3 

59-7 

60.  i 

18000 

60.5 

61.0 

61.4 

61.8 

62.2 

62.7 

63.1 

630 

64.0 

64.4 

19000 

64.8 

65.2 

65-7 

66.1 

66.6 

67.0 

67.4 

67.9 

68.3 

68.7 

2OOOO 

69.2 

69.6 

70.1 

70-5 

71.0 

71.4 

71.9 

72.3 

72.7 

73-2 

2IOOO 

73-6 

74.1 

74.6 

75-0 

75-5 

75-9 

76.4 

76.8 

77-3 

77-7 

22000 

78.2 

78.7 

79.1 

79-6 

80.  i 

80.5 

81.0 

81.5 

81.9 

82.4 

23OOO 

82.9 

83-3 

83.8 

84-3 

84.8 

85-2 

85-7 

86.2 

86.7 

87.1 

24OOO 

87.6 

88.1 

88.6 

89.1 

89.5 

90.0 

90-5 

91.0 

91-5 

92.0 

NOTE.  — In  Table  I  the  scale  of  the  barometer  is  supposed  to 
be  of  brass,  extending  from  the  cistern  to  the  top  of  the  mercu 
rial  column,  and  the  difference  of  expansion  of  brass  and  of  mer 
cury  is  taken  into  account.  The  standard  temperature  of  the 
yard  being  62°  Fahrenheit  and  not  32°,  the  difference  of  expan 
sion  of  the  scale  and  of  the  mercurial  column  carries  the  point 
of  no  correction  down  to  29°  Fahrenheit. 


BAROMETRICAL    MEASUREMENT   OF    HEIGHTS.  171 


ON       i         H    N    N    CO  •*         lOVO  O    C^c 

N  •     •  _;         •   _;  j  -T 


M    N          CO  CO  •*  tOVO          CX 


1X00    0.       ! 


co  10  ix     oo  O  «  co  10   .  ts-oo  0 


5 

oomrowO      vo  ••*•  N.  O  t-x       loroqoO'O 
O   H   w    CO  ro        ^  lOvO    t^  tx      CO   ^  O    O    ••* 


ja     ;      N 


VOCOONO"          N 


CO    O  O    w    N          CO  mvO 


H   c    ro  10VO 


" 


,00         0   Pi   ro  10 


N    -<J-^O  OO    O  <M 


t-»  0    f^O       ^         N    IOCO    H 


•xoiadv    j 


1 
172 

GEODESY. 

TABLE  VII 

—  Giving  the 

Relative  Humidities  in  Fractions  of 

Unity. 

!!* 

Wet  bulb. 

>  —  *> 

*8-o|     h 

10° 

^ 

20° 

22° 

24° 

26° 

28° 

0.0 

I.  000 

I.  OOO 

I.  OOO 

I.  OOO 

i.  ,000 

I.  000    i 

I.  OOO 

.  2 

•925 

.  926 

•  929 

•935  1      .941 

•949 

•  958 

•4 

.854  ; 

.856 

.864 

.874         .887 

.  902 

.919 

.6 

.788  ! 

.791 

.804 

.817         .837 

.858 

.883 

.8 

.727  : 

•  731 

•749 

.764 

.790 

.818 

.850 

I.O 

.671 

.675 

.699 

.717 

•  747 

.781 

.819 

.  2 

.619  j 

.624 

•653 

.  674  I      .  707 

•747  ; 

.791 

•4 

•  570  ; 

•577 

.610 

•635 

.670 

.716 

.765 

.6 

•  525 

,  -534 

•571 

•599 

.636 

.687 

•  741 

.8 

•  483  . 

•494 

•535 

j      -566 

.605 

.660 

.719 

2.0 

•445 

•  456 

.502 

•535 

•577 

•635  ! 

.698 

.2 

.  410 

•  421 

.471 

•  5°6 

•552 

.  612 

.679 

•4 

•377     ; 

•389 

•443 

•479         .529 

•591 

.662 

.6 

•346 

.360 

.417 

•  454         .  507 

•572 

.646 

.8 

•  3i8 

•333 

•393 

•  432         .  487 

•554 

.631 

3-0 

.292 

.308 

•371 

•  412         .469 

.538 

.618 

.2 

.268 

.285 

•351 

•  394         -  453 

•524 

.605 

•4 

•245 

.264 

'    -332 

•377  |      .439 

•593 

.6 

.224 

•  245 

•3*4 

•36i         .426 

•499 

.582 

.8 

.205 

.227 

.298 

•346  |      .413 

.488 

•572 

4.0 

.187 

.  211 

•283 

•332        .401 

.478 

.562 

.  2 

.170 

.  196 

.269 

.  320        .  390 

.469 

•553 

•4 

•154 

.  182 

•257 

•3°9        -379 

.461 

•544 

.6 

.  140 

.I69 

.246 

.299  |      .370 

•453 

.536 

.8 

.  128 

•157 

•  236 

.  290        .  362 

.446 

.528 

5-0 

•  H7 

•  H5 

.226 

•281         .355 

.440 

•  521 

.2 

,      .217 

•273  ;       -348 

•434 

.514 

•4 

.209 

.266         .342 

•429 

•5°7 

.6 

.201 

•259         -336 

.424 

.501 

.8 

.194 

•253 

•331 

.419 

•495 

6.0 

.188 

.248 

•327 

•415 

•489 

.  2 

.182 

•243 

•324 

.411 

.482 

•4 

.I76 

•239 

.321 

.407 

.476 

.6 

•  171 

•  236 

•319, 

•403 

.469 

.8 

.167 

•233 

•3i7 

•399 

•463 

7.0 

1 

.164 

.231 

•315  i 

•396  j 

•457 

BAROMETRICAL    MEASUREMENT    OF    HEIGHTS.              173 

TABLE  VII.  —  Giv&g  the  Relative  Humidities,  &c.  —  Continued. 

11  «• 

1 

Wet  bulb. 

II  £» 

<L  &  <u 

o-e  s 

all 

3°°' 

32° 

34° 

36° 

38° 

40° 

42° 

Q 

o 

I.  OOO 

I.  000 

I.  OOO 

I.  000 

I.  OOO 

I.  OOO 

I.  OOO 

i 

.856 

.885 

.903 

.909 

.914 

.917 

.  920 

2 

.756 

.799 

.821 

.831 

.837 

.843 

.847 

3 

.684 

.728 

.750 

.761 

.768 

.776 

.781 

4 

.628 

.665 

.686 

.697 

.707 

•  715 

.721 

5 

•579 

.609 

.628 

.640 

.652 

.660 

.667 

6 

•533 

.558 

•575 

.589 

.  601 

.610 

.6:8 

7 

.490 

•  512 

.528 

•543 

•554 

.565 

•573 

8 

.450 

.470 

.486 

.  500 

•512 

•  523 

•532 

9 

.414 

.432 

.448 

.  461 

•  473' 

.485 

•495 

10 

•381 

.398 

•413 

.426 

•  438 

.451 

.462 

ii 

•351 

.367 

•381 

•  394 

.406 

.420 

•  432 

12 

.324 

•339 

•  352 

•3^5 

•378 

.391 

.404 

J3 

.299 

•313 

.326 

•339 

•352 

.365 

•378 

14 

.277' 

.289 

.302 

•3r5 

•  328 

.341 

•354 

i5 

•  257 

.268 

.280 

•  293 

.306 

.320 

•332 

16 

.238 

.249 

.261 

.274 

.286 

.300 

.312 

17 

.  221 

.232 

.244 

.256 

.268 

.281 

•  293 

18 

.206 

.216 

.228 

.240 

.252 

.263 

•  275 

^9 

.192 

.202 

.213 

.225 

•  237 

.  246 

.258 

20 

.ISO 

.I89 

.  200 

.211 

.223 

.231 

.242 

21 

.168 

.178 

.189 

.  199 

.209 

.217 

.226 

22 

.158 

.  1  68 

•  178 

.187 

•  195 

.  204 

.213 

23 

.149 

.158 

.167 

•175 

•183 

.192 

.  200 

24 

.140 

.149 

•157 

.164 

.172 

.181 

.188 

25 

.132 

.141 

.148 

•154 

.162 

.170 

.177 

26 

.  1  60 

.168 

27 

•  151 

•  159 

28 

•  143 

•151 

29 

•135 

•  H3 

30 

• 

.128 

•135 

174                                                          GEODESY.                   . 

TABLE  VII.  —  Giving  the  Relative  Humidities,  &c.  —  Continued. 

^  ~  £ 

Wet  bulb. 

g-c  £ 

44° 

46= 

48° 

50° 

52° 

54° 

56° 

0 

o 

I.  OOO 

I.  000 

I.  OOO 

I.  000 

I.  OOO 

I.  000 

I.  OOO 

i 

.922 

•  923 

•  925 

.928 

•93° 

•  931 

•933 

2 

.851 

•  853 

.858 

.862 

.866 

.869 

.872 

3 

.786 

.791 

.796 

.802 

.807 

.811 

.816 

4 

.727 

•  734 

•  740 

•  747 

•753 

.758 

.764 

5 

.674 

.682 

.689 

•  697 

.704 

.709 

.716 

6 

.626 

.635 

.643       .651 

.658 

.664 

.671 

7 

.582 

•592 

.  60  1        .  609 

.616 

.622 

.629 

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•542 

•552 

.562      .570 

•577 

.583 

•590 

9 

.506 

.516 

•  526         .  534 

•541 

•  547 

•554 

10 

•473 

•483 

•493 

.501 

•507 

•513 

•521 

ii 

•443 

•453 

.462 

.470 

.476 

.482 

.491 

12 

•415 

•425 

•433 

.441 

.448 

•  454 

•  463 

J3 

•389 

•398 

.406 

.414 

.422 

.428 

•  437 

14 

•365 

•373 

•  381 

•389 

•397 

.404 

15 

•343 

•35° 

•358 

.366 

•374 

•  382 

•391 

16 

•  322 

•329 

•337 

•345 

•353 

•  361 

•370 

J7 

.302 

.310 

•  318 

•  326 

•333 

•  341 

•350 

18 

.284 

.  292 

.300 

.308 

•315 

.322 

•331 

19 

.267 

•275 

•  283 

.291 

.298 

•305 

20 

•  251 

•259 

.267 

•275 

.282 

.289 

.  296 

21 

.236 

•  244 

.252 

.260 

.267 

.274 

.280 

22 

.  222 

•  230 

•  238 

.246 

•253 

.260 

.265 

23 

.  209 

.217 

.225 

•  233 

.  240 

.246 

.251 

24 

.197 

.205 

.213 

.  221 

.227 

•233 

•  238 

25 

.186 

.194 

.202 

.209 

.215 

.221 

.226 

26 

.176 

.184 

.192 

.  198 

.204 

.  209 

.214 

27 

.167 

•  174 

.182 

.188 

•193 

.198 

.203 

28 

.I58 

.165 

.172 

.178 

.183 

.188 

•193 

29 

'.ISO 

•157 

.163 

.I69 

.174 

.178 

.183 

30 

.  142 

.148 

•154 

.160 

.165 

.  169 

.174 

31 

•151 

.156 

.161 

.166 

32 

•143 

•   .  148 

-153 

.158 

33 

.136 

.141 

.146 

.150 

34 

• 

.129 

•134 

•139 

•  143 

35 

.123 

.127 

•  I32 

.136 

BAROMETRICAL    MEASUREMENT  OF  HEIGHTS.                   175 

TABLE  VII.  —  Giving  the  Relative  Humidities,  &c.  —  Continued. 

II  „• 

Wet  bull). 

>  ^2    <D 

L?   13    IS 

o  ^  £ 

'  SI-SP 

53° 

60° 

62° 

64° 

66°           67°           68° 

0 

o 

I.OOO 

I.OOO 

I.  OOO 

I.  OOO 

I.OOO           I.OOO 

I.OOO 

i 

•935  1 

.936 

.937 

.938 

.940 

.940 

.941 

2 

.875 

.877 

.879 

.881 

.884 

.885 

.886 

3 

.819 

.822 

.825 

.828 

•  832 

•  834 

•835 

4 

.767 

.771 

•775 

•  779 

.783 

.786 

.788 

5 

.719 

.724 

.729 

•734 

•739         .741           -743 

6 

•6/5 

.681 

.686 

.692  1 

.  697  [      .  699  I        .  702 

7 

.634 

.641 

•647         -653 

.  658         .  660           .  663 

8 

•  597  ! 

.603 

.  610         .  616  i 

.621 

.624           .  627 

9 

.562 

.568 

•575 

.582  | 

.587 

•590 

•593 

10 

•  529 

.536 

•543 

•549 

•555 

•5581        -561 

ii 

.499 

.506 

•  513       -519  ; 

.526 

.528 

•531 

12 

•  471 

.478 

•  485 

.491  | 

•  497 

.    .5°° 

•5°3 

13 

•  445 

.452 

•  459 

•  465  i 

.471 

•474 

.476 

H 

.420 

.428 

•434         -440 

.446 

•449 

•  451 

15 

•  397 

.405 

.411         .417 

.422 

.426 

.428 

16 

•  376 

.383 

•  389         •  395 

.  400         .  403 

.  406 

i7 

.356 

.362      .368      .374 

•  379 

•382           .385 

18 

•  337 

.343 

•  348         .  354 

.360 

.3631         -366 

19 

•3*9 

.325 

•  33°         •  336 

•341 

.  344           .  447 

20 

.302 

.308 

•  313         -3*9 

•  324 

.327           .330 

21 

.286 

.292                .297    '           .303 

.308         .311 

•  3T4 

22 

.271 

.2/7               .282                .288 

.  293         .  296 

.299 

23 

.257 

.263 

.268         .273 

.279         .282 

.  284 

24 

.244 

.249 

.  255         •  260 

.265             .268                .2/1 

25 

.231 

.236 

.242  ,      .247 

.252             .255                .258 

26 

.  219 

.  224               .  230                .  235 

.240 

.  243           .  246 

27 

.208 

.213 

.219         .224 

.229       .232  !       .235 

28 

•  .198 

.203 

.208         .213 

.219            .221               .224 

29 

.188 

•193 

.  198         .  203 

.  209 

'.211    i            .214 

30 

.179 

.   184 

.189         .  194 

.200 

.  2O2               .  204 

31 

.170 

•175 

.180 

.185 

.190 

.  193          .  195 

32 

.162 

.167 

,172 

.177 

.182 

.184 

.  ;86 

33 

•  154 

.159 

.  164 

.169 

•173 

.176 

.178 

34 

..147 

.152 

•157 

.161 

.166 

.168 

.170 

35          •  HI 

•145 

.  150         .  154 

•159 

.161 

.163 

I!  " 

i76 


GEODESY. 


TABLE  VII.  —  Giving  the  Relative  Humidities,  &c.  —  Continued. 

li  . 

Wet  bulb. 

IJai 

£  o  "S 

C    r^      g 

^     £"•    r-1 

69° 

70° 

71° 

73°           75° 

77° 

79° 

0 

I.  OOO 

I.  OOO 

I.  OOO 

I.  OOO 

I.  000 

I.  OOO 

I.  OOO 

i           .941 

.942 

.942 

•943         -945 

.946 

.946 

2               .887 

.888 

.889 

.891         .893 

.895 

.896 

3 

.836 

.838 

.839 

•  842         .  845 

.847 

.849 

4 

.789 

.791 

•793 

•  796         .  799 

.801 

.804 

5 

•745 

•747 

•749 

•  753         •  756 

•  759 

.762 

6 

•  7°4 

.706 

.708 

.712         .716 

.719 

.723 

7 

.665 

.668 

.  670 

.674 

.678 

.682 

.686 

8 

.  629 

.632 

•  634 

.638 

.642 

•  647 

.651 

9 

•595 

.598 

.600 

.  605 

.609 

.  614 

.618 

10 

•563 

.566 

.569 

•573 

.578 

•583 

.587 

ii 

•533 

•536 

•539 

•  544 

•549 

•  553 

•558 

12 

.505 

.508 

.511 

.516 

.521 

.526 

•531 

13 

•  479 

.482 

•  484 

.490 

•495 

.  500 

•505 

14 

•  454 

•457 

•  459 

•  465 

.47° 

.476 

.481 

15 

•431 

•434 

•  436 

•  442 

•447 

•453 

•  45s 

16 

.409 

.412 

.414 

.420 

•425 

•  431 

•  436 

17 

•  388 

•391 

•  393 

•399 

•405 

•    .411 

.416 

18 

.368 

•371 

•374 

.380 

•  386 

•  392 

•397 

J9 

•350 

•353 

.356 

•  361 

.367 

•  373 

•379 

20 

•333 

•336 

•339 

•  344 

•350 

•  356 

•  361 

21 

.317 

.320 

•  323 

.328 

•334 

•  340 

•345 

22 

.301 

•304 

.307 

•313 

.319 

•  324 

•33° 

23 

.287 

.  290 

•  293 

•  299 

•  3°4 

.310 

•315 

24 

.274 

.276 

.279 

.285 

.290 

.296 

.301 

25 

.261 

•  263 

.266 

.272 

•  277 

.283 

.288 

26 

.249 

.252 

•  254 

.260 

.265 

.270 

•275 

27 

•  238 

.240 

•  243 

.248 

.-253 

.258 

•  263 

28    . 

.227 

.229 

•  232 

•  237 

.242 

•  247 

•  251 

29 

.217 

.219 

.  222 

.227 

.231 

..236 

.  240 

30 

.  207 

.209 

.212 

.217 

.  221 

.225 

.229 

31 

.  198 

.200 

.  2O2 

.207 

.  211 

.•215 

.      -219 

32 

.  189 

.191 

•193 

.198 

.  2O2 

.205 

.209 

33 

.181 

•I83 

.185 

.  189 

•193 

.  196 

.  200 

34 

•  173 

•175 

.177 

.181 

.  184 

,187 

.191 

35 

.165 

.167 

.  169 

•  173 

.176 

.179 

•  183 

• 

BAROMETRICAL  MEASUREMENT  OF  HEIGHTS.  177 


•&          o 


be 

3 


•£    O  vo    -*•  d    M    O 
£j    t^.  10  ^  ro  01   w 

£ 


y    O    OVOO  \O    10 


^  - 


O    ONCO  ^O    10  ^-  ro  N    M 


w 


^^§ 

G  «-B 


O    ooo   t^.io-^-rorotN    M    o    Oco  oo   Cx'O  MD   LO  ^  TJ-  ro  ro  (M   cl  '"in    w"'' 


b-DH  H  H 

^ 


12 


i78 


GEODESY. 


TABLE  IX. — Second  Part  of  Corrcc 


•^VII 

J  h 

z^  v    V 

H   > 

V. 

(t  +  1') 

(t  +  11) 

180°  F. 

170°  F. 

'160°  F. 

150°  F. 

140°  F. 

130°  F. 

120°   F. 

110°  F.  ' 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

10 

8.5 

7-i 

6.0 

5-0 

4.2 

3-5 

2.9 

20 

16.9 

14-3 

12.0 

10.  0 

8.4 

6.9 

5-8 

30   . 

25.4 

21.4 

iS.o 

15.0 

12.5 

10.4 

8.6 

• 

40 

33-8 

28.5 

23-9 

20.0 

16.7 

13-9 

ii.  5 

50 

42.3 

35-6 

29.9 

25-1 

20.9 

17.4 

14.4 

60 

50-7 

42.8 

35-9 

3O.I 

25-1 

20.8 

17-3 

70 

59-2 

49-9 

41.9 

35-i 

29-3 

24.3 

20.1 

80 

67.6 

57-o 

47-9 

40.1 

33-4 

27.8 

23.0 

9° 

76.1 

64.1 

53-9 

45-i 

37-6 

31.3 

25-9 

100 

84.5 

7i-3 

59-9 

50.1 

41.8 

34-7 

28.8 

no 

93-° 

78.4 

65.8 

55-i 

46.0 

38-2 

31-6 

120 

101.5 

85.5 

71.8 

60.  i 

50.1 

41.7 

34-5 

I30 

109.9 

92.6 

77.8 

65-1 

54-3 

45-1 

37-4 

140 

118.4 

99-8 

83.8 

70.1 

58.5 

48.6 

40-3 

150 

126.8 

106.9 

89.8 

75-2 

62.7 

52.1 

43-1 

160 

135-3 

114.0 

95-8 

80.2 
8-  2 

66.9 

55-6 

46.0 
4.8  0 

170 

1  80 

I43-7 
152.2 

128.3 

107.7 

03.2 

90.2 

71  «o 

.     75-2 

59-O 

62.5 

^vty 

51-8 

1 
190 

160.6 

135-4 

"3-7 

95-2 

79-4 

66.0 

54-6 

200 

169.1 

142.5 

119.7 

100.2 

83.6 

69-5 

57-5 

210 

177.6 

149.6 

125-7 

IOj.2 

87.8 

72.9 

60.4 

22O 

186.0 

156.8 

i3i-7 

110.  2 

91.9 

76.4 

63-3 

230. 

194-5 

163.9 

137-7 

II5-2 

96.1 

79-9 

66.1 

240 

202.9 

171.0 

M3-7 

I2O.2  ' 

100.3 

83-3 

69.0 

250 

211.4 

178.1 

149.6 

125-3 

104.5 

86.8 

71.9 

260 

219.8 

185-3 

155-6 

I30-3 

108.6 

90-3 

74.8 

270 

228.3 

192.4 

161.6 

135-3 

112.  8 

93-8 

77.6 

280 

236.7 

199-5 

167.6 

140.3 

117.0 

97-2 

80.5 

290 

245.2 

206.7 

173-6 

145-3 

121.  2 

100.7 

83-4 

300 

253-6 

213.8 

179.6 

150.3 

125.4 

104.2 

S6.3 

3IO 

262.1 

220.9 

185.6 

155-3 

129.5 

107.6 

89.1 

320 

270.6 

228.0 

191-5 

160.3 

133-7 

in.  i 

92.0 

330 

279.0 

235-2 

197-5 

165.3 

137-9 

114.6 

94-9 

340 

287.5 

242.3 

203.5 

I70-3 

I42.I 

.  118.1. 

97.8 

35° 

295-9 

249.4 

209.5 

I75-4 

146.3 

121.  5 

100.6 

360 

3°4-4 

256.5 

215-5 

180.4 

150.4 

125.0 

103.5 

37° 

312.8 

263-7 

221.5 

185.4 

154-6 

128.5 

106.4 

380 

321-3 

270.8 

227.5 

190.4 

158.8 

132.0 

109.3 

• 

BAROMETRICAL    MEASUREMENT   OF    HEIGHTS.  179 


tlon  for  Atmospheric  Humidity. 


V.  W. 


100°  F. 

90°  F. 

80°  F. 

70°  F. 

60°  F. 

50°  K 

40°  F. 

30°  F. 

20°  F. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

2.4 

1.9 

1.6 

i-3 

i  .  i 

0.8 

0.7 

0.5 

0.4 

4-7 

3-9 

3-2 

2.6 

2.1 

i-7 

i-3 

I.O 

0.8 

7-i 

5-8 

4.8 

3-9 

3-2 

2-5 

2.0 

1.6 

1.3 

9-5 

7-8 

6.4 

5-2 

4.2 

3-4 

2-7 

2.1 

r-7 

11.9 

9-7 

8.0 

6-5 

5-3 

4.2 

3-3 

2.6 

2.  1 

14.2 

11.7 

9.6 

7-8 

6-3 

5-0 

4.0 

3-1 

2-5 

16.6 

13.6 

II  .2 

9.1 

7-4 

5-9 

5-7 

3-7 

2.9 

19.0 

15-6 

12.8 

10.4 

8.4 

6-7 

5-3 

4.2 

3-3 

21.3 

I7-5 

14.4 

xx.  7 

9-5 

7-6 

6.0 

4-7 

3-7 

23-7 

19-5 

16.0 

13-0 

10.5 

8.4 

6.7 

5-2 

4,1 

26.1 

21.4 

17-5 

i4-3 

ii.  6 

9.2 

7-3 

5-8 

4-5 

28.5 

23-4 

19.1 

15-6 

12.6 

10.  I 

8.0 

6-3 

5-0 

30.8 

25-3 

20.7 

16.9 

13-7 

10.9 

8.7 

6.8 

5-4 

33-2 

27-3 

22.3 

18.2 

14.7 

11.  8 

9-3 

7-3 

5-8 

35-6 

29.2 

23-9 

19-5 

15-8 

12.6 

IO.O 

7-9 

6.2 

38.0 

31  .2 

25-5 

20.8 

16.8 

13-4 

10.7 

8.4 

6.6 

40-3 

33-1 

27.1 

22.1 

17.9 

14-3 

"•3 

8.9 

7.0 

•42.7 

35-1 

28.7 

23-4 

18.9 

JS-1 

12.0 

9-4 

7-4 

45-i 

37-0 

30-3 

24.7 

20.  o 

16.0 

12.7 

IO.O 

7-8 

47-4 

39-o 

3L9 

26.O 

21.0 

16.8 

13-3 

IO-5 

8-3 

49-8 

40.9 

33-5 

27-3 

22.1 

17.7 

14.0 

II.  0 

8.7 

52.2 

42.9 

35-1 

23.6 

23.1 

18.5 

14.7 

"•5 

9.1 

54-6 

44-8 

36.7 

29.9 

24.2 

19-3 

15-3 

12.  I 

9-5 

56.9 

46.8 

38.3 

31.2 

2j.2 

20.2 

16.0 

12.6 

9.9 

59-3 

48.7 

39-9 

32.5 

26.3 

21.0 

16.7 

I3-I 

10.3 

61.7 

50.7 

4I'S 

33-8 

27-4 

21.9 

17-3 

13.6 

10.7 

64.0 

52.6 

43-1 

35-1 

28.4 

22.7 

18.0 

14.2 

ii.  i 

66.4 

54-6 

44-7 

36.4 

29o 

23-5 

18.7 

14.7 

xi.  6 

68.8 

56.5 

46.3 

37-7 

30.5 

24.4 

19-3 

15.2 

12.  0 

71.2 

58.5 

47-9 

39-0 

31.6 

25.2 

20.  o 

15-7 

12.4 

73-5 

60.4 

49-5 

40-3 

32.6 

26.  I 

20.7 

I6.3 

12.8 

75-9 

62.4 

51.0 

41.6 

33-7 

26.9 

21.3 

16.8 

I3.2 

78.3 

64-3 

52.6 

42.9 

34-7 

27.7 

22.0 

17-3 

I3.6 

80.7 

66.3 

54-2 

44-2 

35-8 

28.6 

22.7 

17.8 

14.0 

83.0 

68.2 

55-8 

45-5 

36.8 

29.4 

23-3 

18.4 

14.4 

85.4 

70.2 

57-4 

46.8 

37-9 

30-3 

24.0 

18.9 

14.9 

87.8 

72  .  i 

59-o 

48.1 

38.9 

•3".  i 

24.7 

19.4 

r5-3 

90.1 

74.1 

60.6 

49-4 

40.0 

3i-9 

25-3 

19.9 

15.7 

iSo 


GEODESY. 


I 

•<> 


IH 

3   li      IS 

H  H£                                H  M^ 

0 

£ 

1 

§88888888888  888888888888 

ore 

73 

§888888°88888888888  88888 

£ 

| 

§888°8888888  8888888  8888 

C'-f 

•d 

§88°88888  88888  88888  888 

c.'-i- 

? 

§88°8888°888  8888  8888  88 

—  .— 

§  8  °  8  8  8  °  8  8  8  °  8  8  8  °  8  8  8  8  8  8  8  8 

1 

$ 

1 

§  8  888  88  88  888  88  888  8  ; 

o 
o 

-£ 

•d 

§  8  °  8  3  °  8  8  °  8  8  °  8  8  8  8  88  88  8  : 

0 
o 

ich* 

2 

88°8°88  88  8  88  88  8  88  8  \ 

.2 

$ 

1 

§8°  8  8  88  8  88  8  8  88  8  8 

1 

o 

c^'-f 

1 

§808°8°808  88  8  8  8  8  8  8 

o 

S 

1 

§°8°88888888888 

0 
rt 

3! 

^ 

§  °  8  °  8  °  8  °  8  °  8  °  8  8  8  8  8  8  § 

0 

* 

-a 

§  °  §  °  8  °  °  8  °  8  °  8  °  °  8  °  8  °  °  8  °  8  °  8  § 

"o 

_'-*• 

5 

§  °  8  °  °  8  °  8  °  8   88   8   8  8  § 

o 

6 

a 

oofe 

•a 

§  °  8  °  °  8  °  °  8  °  °  8  °  °  8  °  °  8  °  °  8  °  °  8  § 

«J 

'-f 

3 

0  0  H  0  0  0  g  00  «  0  0  0  H  0  0  M  0  0  H  0  0  0  H  g   | 

oo      oo      ooq      q  q   , 

H5 

1 

googooogooogooogooogo   o  g  g 

"£ 

T3 

|  00  g  0000  g  0000  go   OgOOOOgOg 

* 

1 

§000"00000w00000^00000w0g 
8         8         o        q   q 

"ft 

•d 

13 

§OOOOwOOOOOOOMOOOOOO«OOOQ 
8             8           q      q 

•* 

•d 

T3 

§OOOOOOHOOOOOOOOOOOgOOOOQ   | 
8                q      o 

_}* 

| 

§000000000000"00'  000000000 
8             q 

o 

i    0  H  N  m  -*  l^\0  t^CO  0  0  H  0  ro  -*•  >0.0  t-03  ON 

t^OO   &  O   "         H    M 


THERMOMETRICAL    MEASUREMENT    OF    HEIGHTS. 


181 


XL VI. — TJiermometrical  Measurement  of  Heights. 

BAROMETRIC     PRESSURES     CORRESPONDING     TO     TEMPERATURES     OF      THE 
BOILING-POINT    OF   WATER. 


o  *: 
°o 
in   '% 

Tenths  of  a  degree  of  Fahrenheit. 

|| 

0 

2 

4 

6 

8 

I85 

17.048 

17.  122 

17.  197 

17.272 

17.348 

1  86 

•  423 

•499 

•  575 

.652 

.729 

187 

.806 

.883. 

.961 

18.039 

18.117 

1  88 

IS.  195 

18.274 

iS.353 

•  432 

•512 

189 

•592 

.672 

•753 

.833 

.914 

190 

.996- 

19.077 

I9.I59 

.19-  241 

19-  324 

191 

19.407 

.490 

r**-. 

.657 

.741 

192 

.825 

.910 

•995 

20.  080 

20.  1  66 

193 

20.  251 

20.  338                 20.  424 

.5II                       .598 

194 

.685 

•  773                -861 

.949                 21.038 

195 

21.  126 

21.  2l6 

21.305 

21.395                       .485 

196 

.576 

.  666                 .  758 

.  849                       .  941 

197 

22.  033 

22.  125 

22.  2l8 

22.311                  22.404 

198 

.498 

•592 

.686 

.781                       .876 

199 

.971 

23.067      i           23.163 

23.259                 23.356 

200 

23.  453 

•550 

.648 

.746      |                 .845 

2OI 

•943 

24.  042    •             24.  142 

24.  241                  24.  341 

2O2 

24.442 

.542 

.644 

•  745     i            •  847 

203 

-.949 

25.051 

25.154 

25-257            25.361 

204 

25.  465 

.569 

.674 

.  779                .  884 

.       205 

.990 

26.  096 

26.  202 

26.309            26.416 

2O6 

26.523 

.63I 

.  740 

.848 

•  957 

207 

27.066 

27.176                 27.286 

27.397 

27-507 

208 

.618 

.730      i                 .842 

•954 

28.  067 

209 

28.  1  80 

28.293                 28.407 

28.521 

.636 

210 

.751 

.866 

.982 

29.  098 

29.215 

211 

29-331 

29.  449 

29.  566 

.684 

.803 

212 

.922 

30.  041 

30.  161 

30.  281 

30.  401 

182 


GEODESY. 


Table  of  Comparison  of  Fahrenheit's  Thermometer  with  Reaumur's 
and  tJie  Centesimal. 


Fah. 

Reaum. 

Centes. 

Fah. 

Reaum. 

Centes. 

Fah. 

Reaum. 

Centes. 

0 

0 

0 

o 

o 

0 

0 

0 

0 

33 

+  0.4 

+  0.6 

67 

+  15-6 

+19.4 

O 

—14.2 

-17.8 

34 

0.9 

i.  i 

68 

16.  o 

20.  0 

I 

13.8 

17.2 

35 

1.3 

1-7 

69 

16.4 

20.  6 

2 

13.3 

16.7 

36 

1.8 

2.2 

70 

16.9 

21.  I 

3 

12.9 

1  6.  i 

37 

2.  2 

2.8 

7i 

J7.3 

21.  7 

4 

12.4 

15.6 

38 

2-7 

3-3 

72 

17.8 

22.2 

5 

12.0 

•15.0 

39 

3-1 

3-9 

73 

18.2 

22.8 

6 

ii.  6 

14.4 

40 

3-6 

4.4 

74 

18.7 

23-3 

7 

ii.  i 

13-9 

4i 

4.0 

5-0 

75 

19.1 

23-9 

8 

10.7 

13.3 

42 

4.4 

5-6 

76 

19.6 

24.4 

9 

IO.  2 

12.8 

43 

4-9 

6.1 

77 

20.  o 

2$.  O 

10 

9.8 

12.  2 

44 

5-3 

6-7 

78 

20.  4 

25.6 

ii 

9-3 

11,7 

45 

5.8 

7.2 

79 

20.9 

26.1 

12 

8.9 

ii.  i 

46 

6.2 

7.8 

'So 

21.3 

26.7 

13 

8.4 

10.  6 

47 

6.7 

8-3 

Si 

21.8 

27.2 

14 

8.0 

IO.  O 

48 

7-i 

8.9 

82 

22.2 

27.8 

11 

7.6 
7-  i 

9-4 
8.9 

49 
50 

7.6 
8.0 

9-4 

IO.  O 

83 
84 

22.7 
23.1 

28.3 
28.9 

17 

6-7 

8'3  : 

5i 

8.4 

10.  6 

85 

23.6 

29.4 

18 

6.2 

7-8 

52 

8.9 

.     ii.  i 

86 

24.  o 

3o.-o 

J9 

5.8 

7.2 

53 

9-3 

11.7 

87 

24.4 

30.6 

20 

5-3 

6-7 

54 

9.8 

12.2 

88 

24.9 

3I.I 

21 

4.9 

6.1  i 

55 

10.  2 

12.8 

89 

25-3 

31-7 

22 

4-4 

5-6  ; 

56 

10.7' 

13-3 

90 

25.8 

32.2 

23 

4.0 

5-o  , 

57 

II.  I 

13-9 

9i 

26.2 

">-7     8 

02.  5 

24 

3-6 

4-4  ! 

58 

ii.  6 

14.4 

92 

26.7 

33-3 

25 

3-  ! 

3-9 

59 

12.  O 

15.0 

93 

27.1 

33-9 

26 

2.7 

3-3 

60 

12.4 

I5.6 

94 

27.6 

34-4 

27 

2    2 

2.8 

61 

12.9 

16.  i  | 

95 

28.0 

35-0 

28 

1.8 

2.2 

62 

13.3 

16.7 

96 

28.4 

35-6 

29 

1-3 

i-7 

63 

13-8 

17.2 

97 

28.9 

36.1 

30 

0.9 

1.  1 

64 

14.2 

17.8  ; 

98 

29.3 

36.7 

31 

32 

—   0.4 
O.  O 

—  o.  6 

0.0 

3 

14.7 
+  I5.I 

18.3 

+i8.9 

99 

IOO 

29.8 

+30.2 

+?:5 

x°  Reaumur  =  (32°  +  $x°)  Fah.  =  -|  x°  Centes. 
x°  Centes.  =  (32°  +  f-  x°)  Fah.  =  f  x°  Reaum. 
x°  Fah.  =  J  (x°  —  32°)  Reau.  =  £  (.r°  —  32°)  Cen. 


TABLES    AND     FORMULAE. 


PART     III. 


ASTRONOMY. 


ASTRONOMY. 


XLVIL—  Of  Sidereal  and  Solar  Time. 

True  or  apparent  solar  time  is  that  deduced  from  observations 
of  the  sun,  or  is  the  same  as  that  shown  by  a  well-adjusted  sun 
dial. 

Mean  solar  time  is  derived  from  the  time  employed  by  the 
earth  in  revolving  on  its  axis,  as  compared  with  the  sun,  supposed 
to  move  at  a  mean  rate  in  its  orbit,  and  to  make  365.242218 
revolutions  in  a  mean  Gregorian  year. 

It  cannot  be  immediately  obtained  from  observation,  but  is  al 
ways  deduced  from  apparent  time  by  the  aid  of  the  equation 
of  time,  which  is  the  angular  distance,  in  time,  between  the  mean 
and  true  sun;  or,  mean  solar  time  =  apparent  solar  time  i  equa 
tion  of  time. 

Sidereal  time  is  the  portion  of  a  sidereal  day  which  has  elapsed 
since  the  transit  of  the  first  point  of  Aries. 

Its  point  of  origin  cannot  be  determined  by  observation,  but 
it  is  known  at  any  moment  by  the  right  ascension  of  whatever 
object  may  be  then  in  the  meridian  ;  or, 

Sidereal  time  of  a  star's  culmination  =  AR.  of  >K  ; 

Sidereal  time  at  mean  noon  =  AR.  mean  %  at  mean  noon ; 
and,  generally, 

Sidereal  time  =  sidereal  time  at  mean  noon  i  solar  time  from 
mean  noon,  (expressed  in  sidereal  intervals;) 

Solar  time  =  sidereal  time  —  sidereal  time  at  mean  noon,  (the 
difference  being  reduced  to  a  solar  interval.) 


1 86  ASTRONOMY. 


XL VI I. — Of  Sidereal  and  Solar  Time — Continued. 

EXAMPLE. 

To  find  the  mean  solar  time  of  the  passage  of  Altair  over  the 
meridian  of  Washington,  on  the  loth  July,  1849: 

h.   m.     s. 
AR.  Altair,  July  10,  1849 =  19  43  27.39 

Sidereal  time  at  mean  noon  at  Washington —    714  00.96 

Sidereal  interval  past  Washington  mean  noon. . .  =  12  29  26.43 
Retardation  of  mean  on  sidereal  time =  —  02  02.77 


Corresponding   mean    time    interval   past   mean 

noon  or  mean  time  of  culmination =  12  27  23.66 

The  nautical  almanacs  give  the  sidereal  time  at  mean  noon 
for  each  day  of  the  year  for  a  certain  meridian. 

If  the  sidereal  day  be  taken  equal  to  24  sidereal  hours,  the 
mean  solar  day  will  be  equal  to  2411  3m  56s. 5 5  of  those  sidereal 
hours ;  or  the  daily  acceleration  of  sidereal  on  mean  solar  time 
(which  is  the  mean  motion  of  the  earth  in  a  mean  solar  day)  is 
3m  56s. 5 5 54  of  sidereal  time;  hence  the  sidereal  time  at  mean 
noon  under  any  meridian  other  than  that  of  the  nautical  alma 
nac  used  \vill  be  found  by  allowing  the  proportion  of  this  quan 
tity  due  to  the  difference  of  longitude  of  the  two  places. 

If  the  mean  solar  day  be  taken  equal  to  24  mean  solar  hours, 
the  sidereal  day  will  be  equal  to  23^  56m  4s.o9  of  those  solar 
hours,  or  the  daily  retardation  of  mean  solar  on  sidereal  time  is 
3m55s-9°93  °f  s°lar  time. 

The  astronomical  day  begins  at  noon.  In  the  civil  or  common 
method  of  reckoning,  the  day  is  supposed  to  commence  at  the 
preceding  midnight.  The  civil  reckoning  is  therefore  1 2  hours  in 
advance  of  the  astronomical  reckoning,  and  in  the  above  example, 
July  loth,  i2h  27m  23^.66  astronomical  time,  corresponds  to  July 
nth,  ol1  27m  23s. 66  a.  m.  civil  time. 


TIME.  187 


XL VI II. —  To  find  the    Time  by  an  Altitude  of  the    Sun  or  a 

Star. 


Sidereal  time  =  AR.  X  ±  *  's  hour-angle. 
Solar  time       =  24h        =J=  &'s  hour-angle. 
2  m  =  L  +  A  +  A 


cos  L  ."sin  A 

where — 

L  =  the  latitude  of  the  place  of  observation  ; 

A  =  the. north  polar  distance  of  the  sun  or  the  star; 

A  =  the  corrected  altitude  of  the  sun  or  star 

=  observed  altitude  —  (refraction  —  parallax)  i  semi-di 
ameter;  and 
p  =  the  'hour-angle  of  the  sun  or  star. 

The  formula  gives  the  arc  in  degrees,  which  must  be  converted 
into  time,  as  in  one  of  the  following  four  cases: 

1.  When  we  have  the  corrected  altitude  of  the  sun's  center,  the 
hour-angle,  /,  in  time,  is  the  apparent  time  when  the  sun  is  in  the 
west,  or  the  complement   of   24  hours  when  in  the   east.     To 
reduce  it  to  mean  time  apply  the  equation  of  time. 

2.  But  should  the  sidereal  time  be    required,   transform   the 
mean  time  thus  obtained  to  sidereal  time,  as  previously  explained. 

3.  When  the  altitude  is  that  of  a  star,  the  sidereal  time  is  at 
once  deduced  from  the  hour-angle,/. 

4.  And  if,  in  this  last  instance,  solar  time  should  be  required, 
convert  this  sidereal  time  into  solar  time  by  means  of  the  equa 
tion — 

Solar  time  =  AR.  *  —  AR.  € 


in  which  the  sign  -f-  is  used  if  the  star  is  observed  in  the  west, 
and  the  sign  —  if  in  the  east ;  or, 

Mean  solar  time  =  the  mean  solar  equivalent  of  (sidereal  time 
of  observation  —  sidereal  time  of  preceding  mean  noon  at  place.) 


i88 


ASTRONOMY. 


The  most  common  dial  is  that  in  which  the  plane  of  the  dial  is 
horizontal,  and  the  style,  placed  in  the  meridian,  is  inclined  to  the 
plane  of  the  dial  at  an  angle  equal  to  the  latitude  of  the  place. 

Hour-lines  are  drawn  from  the  center,  or  point  where  the  style 
intersects  the  plane,  to  the  exterior  limit  of  the  surface  of  the 
dial.  Their  positions  are  calculated  from  the  formula — 

tan  x  =  tan  p  sm  L 
in  which — 

x  =  hour-angle  on  the  horizontal  plane ; 

/=  15°.  30°,  45°,  etc.,  the   hour-angle    on  the    equatorial 

plane  \  and 

L  =  latitude  of  the  place. 

The  geometrical  determination  of  these  lines  will  be  readily 
seen  from  the  following  figure. 


\ 


SUN-DIAL. 


189 


.—  Sun-  Dial—  Continued. 


As  the  lines  of  IV,  V,  VIII,  and  VII  cannot,  generally,  be 
directly  drawn,  owing  to  want  of  space  upon  the  surface  of  the 
dial,  draw,  from  any  point  of  the  line  of  IX  hours,  a  line  parallel 
to  that  of  III  hours,  and  take  ab1  '  =  ab,  and  ac1  =  ac\  b1  and  c 
will  be  points  in  the  lines  VIII  and  VII.  The  lines  IV  and  V 
will  make  the  same  angles  on  the  opposite  side.  The  line  of  VI 
is  perpendicular  to  that  of  XII. 

To  determine  the  Meridian  Line. 

Take  a  point  in  the  plane  of  the  dial  through  which  it  is  in 
tended  the  meridian  plane  shall  pass.  With  this  point  as  a  cen 
ter  describe  several  concentric  circles.  Fix  a  straight  pin  in  the 
center,  perpendicular  to  the  plane  of  the  dial,  of  such  a  length 
that  the  extremity  of  the  shadow  cast  by  it  shall  fall  within  the 
circles  at  XII  M.  Mark  the  points  where  the  extremity  of  the 
shadow  passes  over  these  circles  in  the  forenoon  and  again  the 
same  in  tfye  afternoon.  Ihe  line  drawn  from  the  middle  of  these 
arcs,  contained  between  the  points  of  passage,  to  the  center  of 
the  circles  will  be  the  meridian. 


SUN-DIAL   CORRECTION. 

Mean  Time  at  Apparent  Noon. 


1 

Day.     January. 

February. 

March. 

April. 

May. 

June. 

. 

//.      m. 

//.     ;;/.         //.     m. 

//.     Di. 

h.     DI. 

h.     DI. 

I             12        4 

12     14 

12       12 

12       4 

ii     57 

ii     58 

8             12         7            12       14             12       II 

12         2 

ii     56 

ii     59 

l6             12       10             12       14      1       12         9 

12         O 

ii     56 

12        O 

24             12       12 

12       I3 

12         6 

II       58 

ii-  57 

12         2 

Day. 

July. 

August. 

September. 

October. 

November 

December. 

! 

h      m 

//  .     111 

Ji.     m. 

h.      in.          h.      DI. 

//.     ;;/. 

I 

12      3 

12         6 

12         0 

II     50          ii     44 

II     50 

8 

12      5 

12         £ 

II       58 

II     48          II     44 

ii     53 

16 

12      .   6 

12      4 

"    55 

ii     46          ii     45 

ii     56 

24 

12         6 

12         2 

ii    52 

ii     45         ii     47 

12         0 

1 

. 

1  90                                                         ASTRONOMY. 

For  converting  Intervals  of  SIDEREAL  into  Corresponding  Intervals 

of  MEAN  SOLAR  Time. 

Hours. 

Minutes. 

Seconds. 

h. 

in,     s. 

m. 

S. 

m. 

s. 

s. 

S. 

s. 

s. 

i 

o     9.830 

i 

0.164 

3i 

5-079 

i 

0.003 

31 

0.085 

2 

o  19.659 

2 

0.328 

32 

5-242 

2 

0.005 

32 

0.087 

3 

o  29.489 

3 

0.491 

33 

5.406 

3 

O.OO8 

33 

0.090 

4 

o  39,318 

4 

0.655 

34 

5-570 

4 

O.OII 

34 

0.093 

5 

o  49.148 

5 

0.819 

35 

5-734 

5 

0.014 

35 

0.096 

6 

o  58.977 

6 

0.983 

36 

5.898 

6 

0.016 

36 

0.098 

7 

i     8.807 

7 

1.147 

37 

6.062 

7 

0.019 

37 

O.  IOI 

8 

'  i   18.636 

8 

1.311 

38 

6.225 

8 

O.O22 

38 

0.104 

9 

i  28.466 

9 

1.474 

39 

6.389 

9 

O.O25 

39 

o.  1  06 

10 

i  38.296 

10 

1.638 

40 

6.553 

10 

0.027 

40 

o.  109 

ii 

i  48.125 

ii 

1.802 

4i 

6.717 

ii 

0.030 

4i 

O.  112 

12 

i  57-955 

12 

1.966 

42 

6.881 

12 

0.033 

^2 

O.II5 

13 

2       7.784 

13 

2.130 

43 

7.044 

13 

0.036 

43 

O.IlS 

T4 

2    17.614 

14 

2.294 

44 

7.208 

14 

0.038 

4-4 

O.  1  2O 

15 

2    27.443 

15 

2.457 

45 

7-372 

15 

0.041 

45 

o.  123 

16 

2    37.273 

16 

2.621 

46 

7.536 

16 

0.044 

46 

o.  126 

17 

2    47.103 

17 

2.785 

47 

7.700 

17 

0.047 

47 

0.128 

18 

2    56.932 

18 

2.949 

48 

7-864 

18 

0.049 

48 

0.131 

19 

3     6.762 

T9 

3-  "3 

49 

8.027 

19 

0.052 

49 

0.134 

20 

3  I6.591 

20 

3-277 

50 

8.191 

20 

0.055 

50 

0.137 

21 

3  26.421 

21 

3.440 

5i 

8.355 

21 

0.057 

5i 

o.  140 

22 

3  36.250 

22 

3.604 

52 

8.519 

22 

O.O60 

52 

0.142 

23 

3  46.080 

23 

3.768 

53 

8.683 

23 

0.063 

53 

0.145 

24 

3  55.909 

24 

3-932 

54 

8.847 

24 

O.O66 

54 

0.148 

25 

4.096 

55 

9.010 

25 

0.068 

55 

0.150 

26 

4-259 

56 

9.174 

26 

O.O7I 

56 

0.153 

27 

4-423 

57 

9-338 

27 

0.074 

57 

0.156 

28 

4.587 

58 

9.502 

28 

0.076 

58 

0.159 

29 

4.751 

59 

9.666 

29 

o  .  079 

59 

o.  161 

30 

4.915 

60 

9.830 

30 

0.082 

60 

0.164 

The  quantities  taken  from  this  table  must  be  subtracted  from  a  sidereal  interval  to 
obtain  the  corresponding  interval  in  mean  solar  time. 

TIME. 


For  converting  Intervals  of  MEAN  SOLAR  Time  into  Corresponding 

Intervals  of  SIDEREAL  Time. 

Hours. 

Minutes. 

Seconds.  . 

h. 

m.     s. 

tn. 

S. 

<jn. 

j. 

s. 

S. 

s. 

s. 

i 

o    9.856 

i 

0.164 

3i 

5.092 

i 

0.003 

31 

0.085 

2 

o  19.713 

2 

0.329 

32 

5-257 

o 

O.OO5 

32 

o.oSS 

3 

o  29.  569 

3 

0.493 

33 

5.421 

3 

0.008 

33 

0.090 

4 

o  39.426 

4 

0.657 

34 

5.585 

4 

O.OII 

34 

0.093 

5 

o  49.282 

5 

0.821 

35 

5-750 

5 

0.014 

35 

0.096 

6 

o  59.!39 

6 

0.986 

36 

5.9M 

6 

0.016 

36 

0.098 

7 

i     8.995 

7' 

1.150 

37 

'6.078 

7 

0.019- 

37 

O.  IOI 

8 

i  18.852 

8 

I.3I4 

38 

6.242 

S 

0.022 

38 

0.104 

9 

i  28.708 

9 

1.478 

39 

6.407 

9 

0.025 

39 

0.106 

10 

i  38.565 

10 

1.643 

40 

6.571 

10 

0.027 

40 

0.109 

ii 

i  48.421 

ii 

1.807 

4i 

6.735 

ii 

0.030 

4i 

O.II2 

12 

i  58*278 

12 

1.971 

42 

6.900 

12 

0.033 

42 

o.  115 

13 

2      8.134 

13 

2.  136 

43 

7.064 

13 

0.036 

43 

0.118 

14 

2    I7-991 

14 

2.300 

44 

7.228 

14 

0.038 

44 

O.T20 

15          2    27.847 

15 

2.464 

45 

7.392 

15 

0.041 

45 

0.123 

16 

2    37.704 

16 

2.628 

46 

7-557 

16 

0.044 

46 

o.  126 

17 

2    47-560 

17 

2-793 

47 

7.721 

17 

0.047 

47 

0.129 

18 

2    57.416 

18 

2-957 

48 

7-885 

iS 

0.049 

48 

0.131 

19 

3     7-273 

19 

3-  121 

49 

8.050 

19 

0.052 

49 

0.134 

20 

3  17-129 

20 

3.285 

50 

8.214 

20 

0.055 

50 

0.137 

21 

3  26.986 

21 

3-450 

51 

8.378 

21 

0.057 

51 

o.  140 

22 

3  36.842 

22 

3.614 

52 

8-542 

22 

O.O6O 

52 

0.142 

23 

3  46.699 

23 

3-7/8 

53 

8.707 

23 

0.063 

53 

0.145 

24 

3  56.555 

24 

3-943 

54 

8.871 

24 

O.O66 

54 

0.148 

25 

4.107 

55 

9-°35 

25 

0.068 

55 

0.151 

26 

4.271 

56 

9.199 

26 

0.071 

56 

0.153 

27 

4.436 

57 

9-364 

27 

0.074 

57 

o.  156 

23 

4.600 

58 

9-528 

28 

0.077 

58 

0.159 

29 

4.764 

59 

9.692 

29 

0.079 

59 

o.  161 

. 

30 

4.928 

60 

9-856 

30 

0.082 

60 

o.  164 

The  quantities  taken  from  this  table  must  be  added  to  a  mean  interval  to  obtain  the 

corresponding  interval  in  sidereal  time. 

I92 


ASTRONOMY. 


To  convert  Parts  of  the  Equator  in  Arc  into  Sidereal  Time,  or  to 
convert  terrestrial  Longitude  in  Arc  into  Time. 


Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

! 

Time. 

° 

h.  111. 

o 

h.  m. 

0 

//.  m. 

//.   7/7. 

I 

o   4 

31 

2   4 

61 

4   4 

91 

6   4 

2 

o   8 

32 

2    8 

62 

4   8 

92 

6   S 

3 

0   12 

33 

2   12 

63 

4  12 

93 

6   12 

4 

o  16 

. 

34 

2   l6 

64 

4  16 

94 

6  16 

5 

0   20 

35 

2   20 

65 

4  20 

95 

6  20 

6 

o  24 

;  36 

2   24 

66 

4  24 

96 

6  24 

7 

o  28 

37 

2   28 

67 

4  2S 

97 

6  28 

8 

o  32 

33 

2   32 

68 

4  32 

98 

6  32 

9 

o  36 

'  39 

2   36 

69 

4  36 

99 

6  36 

10 

o  40 

!  4° 

2   40 

70 

4  40 

100 

6  40 

ii 

o  44 

41 

2  44 

71 

4  44 

101 

6  44 

12 

o  48 

42 

2   48 

72 

4  48 

102 

6  48 

13 

o  52 

43 

2   52 

73 

4  52 

103 

6  52 

M 

o  56 

44 

2   56 

74 

4  56 

104 

6  56 

15 

I    0 

45 

3   o 

75 

5   o 

105 

7   o 

16 

i   4 

46 

3   4 

76 

5   4 

1  06 

7   4 

17 

i   8 

47 

3   8 

77 

5   8 

107 

7   8 

18 

I   12 

48 

3  12 

78 

5  12 

108 

7  12 

IQ 

i  16 

49 

3  16 

79 

5  16 

109 

7  16 

20 

I  20 

50 

3  20 

80 

5  20 

no 

7  20 

21 

I  24 

5* 

3  24 

Si 

5  24 

III 

7  24 

22 

I   28 

52 

3  28 

82 

5  28 

112 

7  28 

23 

I  32 

53 

3  32 

83 

5  32 

113 

7  32 

24 

i  36  ! 

54 

3  36 

84 

5  36 

114 

7  36 

25 

I  40 

55 

3  40 

85 

5  40 

H5 

7  40 

26 

i  44 

56 

3  44 

86 

5  44 

116 

7  44 

27 

i  48 

57 

3  48 

87 

5  48 

117 

7  48 

28 

i  52 

58 

3  52 

88 

5  52 

118 

7  52 

29 

i  56 

59 

3  56 

89 

5  56 

119 

7  56 

30 

2    O 

60 

4   o 

go 

6   o 

I2O 

8   o 

SPACE  INTO  TIME.                     193 

To  convert  Parts  of  the  Equator  in  Arc  into  Sidereal  Time,  or  to 
convert  7errestrial  Longitude  in  Arc  into  Time  —  Continued. 

DEGREES. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

0 

h.  m. 

0 

h.  tn. 

0 

h.  in. 

° 

h.  DI. 

121 

8   4 

151 

10   4 

181 

12    4 

211 

14   4 

122 

8   8 

152 

10   8 

182 

12    8 

212 

14   8 

123 

8   12 

153 

10   12 

183 

12   12 

213 

14   12 

124 

8  16 

154 

10  16 

184 

12   l6 

214 

14  16 

125 

8   20 

155 

10   20 

185 

12   2O 

215 

14   20 

126 

8  24 

156 

10  24 

1  86 

12   24 

216 

14   24 

127 

8  28 

157 

10   28 

187 

12   28 

217 

14   28 

128    8  32 

158 

10  32 

iSS 

12   32 

218 

14   32 

129 

S  36 

159 

10  36 

189 

12   36 

219 

14   S^ 

,30 

131 
132 

8  40 

8  44 

8  48 

160 

161 

162 

10  40 

10  44 
10  48 

190    12   40 

191    12  44 

192    12   48 

22O 

|   221 
222 

14   40 

14  44 
14  48 

133 

8  52 

163 

10  52 

193.  1   12   52 

223 

14  52 

134 

8  56 

164 

10  56 

194    12   56 

224 

M  S^ 

135 

9   o 

165 

II    0 

195   13   o    225 

15    0 

I36 

9   4 

1  66 

ii   4 

196 

13   4 

1   226 

15   4 

137 

9   8 

167 

IT   8 

197 

13   8 

227 

15   8 

133 

9  12 

168 

II   12 

I98 

13   12 

228 

15   12 

139 

9  16 

169 

ii  16 

199 

13  16 

229 

15  16 

140 

9  20 

170 

II   20 

2OO 

13  20 

230 

15  20 

141 

9  24 

171 

II   24 

201 

13  24 

231 

15  24 

142 

9  28 

172 

II   28 

202 

13  28 

232 

15  28 

143 

9  32 

173 

II   32 

203 

13  32 

233 

15  32 

144 

9  36 

174 

II   36 

2O4 

13  36 

234 

15  36 

145 

9  .  40 

175 

II   40 

205 

13  40 

235 

15  40 

I46 

9  4-1 

176 

ii  44 

206 

13  44 

236 

15  44 

147 

9  48 

177 

ii  48 

207 

13  48 

237 

15  48 

148 

9  52 

178 

ii  52 

208 

13  52 

238 

15  52- 

149 

9  56 

179 

ii  56 

209 

13  56 

239 

15  56 

150 

10    0 

1  80 

12    O 

2IO 

14   o 

240 

16   o 

1  94                       ASTRONOMY. 

To  convert  Parts  of  the  Equator  in  Arc 

into  Sidereal  Time,  or  to 

convert  Terrestrial  Longitude  in  Arc  into  Time  —  Continued. 

DEGREES. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

o 

//.  m. 

0 

h.  in. 

° 

//.   ;;/. 

° 

h.  m. 

241 

16   4 

271 

18   4 

301 

20   4 

331 

22    4 

242 

16   8 

272 

18   8 

302 

20    8 

332 

22    8 

243 

16  12 

273 

IS   12 

303 

20   12 

333 

22   12 

244 

16  16 

274 

18  16 

304 

20   16 

334 

22   l6 

245 

16  20 

275 

IS   20 

305 

2O   2O 

335 

22   20 

246 

16  24 

276 

18  24 

306 

20   24 

336 

22   24 

247 

16  28 

277 

18  28 

307 

2O   28 

337 

22   28 

24S 

16  32  '   2.78 

18  32 

!  308 

20   32 

333 

22   32 

249 

16  36    279 

18  36 

309 

20   36 

339 

22   36 

250   16  40 

280 

1  8  40 

1  310 

20   40 

340 

22   40 

251 

16  44 

281 

18  44 

!   3I1 

20  44 

34i 

22  44 

252 

16  48  L  282 

18  48 

:  312 

20   48 

342 

22   48 

253 

16  52    283 

ia  52 

313 

20   52 

343 

22   52 

254 

16  56    284 

18  56 

3i4 

20  56    344 

22   56 

255      17     0    !    2S5 

19   o 

3i5 

21   o    345    23   o 

256  I  17   4    286 

19   4    316 

21   4 

346 

23    4 

257  ;  17   8    287 

19   8   ;  317 

21    S 

347 

23   8 

258    17   12      288 

19   12   li   318 

21   12 

348 

23   12 

259   17  16 

!  289 

19  16 

319 

21   l6 

349 

23  16 

260  ;  17  20 

290 

19   20 

320 

21   2O 

350 

23   20 

261   17  24 

291 

I9  •  24 

321 

21   24 

35i 

23   24 

262   17  28 

292 

19   28 

322 

21   28 

352 

23  28 

263 

17  32 

293 

19   32 

323 

21   32 

353 

23  32 

264   17  36 

294 

19   36 

324 

21   36 

354 

23  36 

265   17  40 

295 

19   40 

325 

21   40 

355 

23  40 

266 

17  44 

296 

19  44 

326 

21  44 

356 

23  44 

267 

17  48 

297 

19  48 

327 

21   48 

357 

23  48 

268 

17  52 

298 

19  52 

328 

21   52 

358 

•  23  52 

269 

17  56 

299 

19  56 

329 

21   56 

359 

23  56 

270 

18   o 

300 

20    0   i   330 

22    0 

360 

24   o 

SPACE    INTO    TIME. 


To  convert  Parts  of  the  Equator  in  Arc  into  Sidereal  Time,  or  to 
convert  Terrestrial  Longitude  in  Arc  into  Time — Continued. 


MINUTES. 

SECONDS. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

' 

in.     s. 

' 

m.     s. 

" 

s. 

" 

s. 

i 

o      4 

3i 

2       4 

i 

0.067 

3i 

2.067 

2 

o       8 

32 

2         S 

2 

0.133 

32 

2.133 

3 

0      12 

33 

2       12 

3 

0.200 

33 

2.200 

4 

o     16 

34 

2       l6 

4 

0.267 

34 

2.267 

5 

o     20 

35 

2      2O 

5 

0-333 

35 

2-333 

6 

o    24 

36 

2      24 

6 

O.40O 

36 

2.400 

7 

o     28 

37 

2      28 

7 

0.467 

37 

2.467 

8 

o    32 

33 

2      32 

8 

0-533 

38 

2-533 

9 

o    36 

39 

2      36 

9 

0.600 

39 

2  .  6()O 

10 

o    40 

40 

2      40 

10 

0.667 

40 

2.667 

ii 

o     44 

4i 

2     44 

ii 

0.733 

41 

2-733 

12 

o     48 

42 

2      48 

12 

0.800 

42 

2.800 

13 

o     52 

43 

2       52 

13 

0.867 

43 

2.867 

14 

o     56 

44 

2       56 

14 

0-933 

44 

2-933 

15 

I         0 

45 

3      o 

15 

I.  000 

45 

3.000 

16 
17 

i       4 

I       8 

46 

47 

3       4 

3       8 

16 

17 

1.067 

1.133 

46 
47 

3.067 
3.133 

18 

I       12 

48 

3     12 

18 

I.2OO 

48 

3.20O 

19 

i     16 

49 

3     16 

IQ 

1.267 

49 

3.267 

20 

I       20 

50 

3     20 

20 

1-333 

50 

3o33 

21 

I       24 

51 

3     24 

21 

1.400 

5i 

3.400 

22 

I       28 

52 

3     28 

22 

1.467 

52 

3.467 

23 

I       32 

53           3     32 

23 

1-533 

53 

3-533 

24 

I       36 

54 

3     36 

24 

i  .  600 

54 

3  -  600 

25 

I      40 

55 

3     40 

25 

1.667 

55 

3.667 

26 

27 

i     44 

i     48 

56 

57 

3     44 
3     48 

26 

27 

1-733 
i  .  800 

56 

57 

3-733 
3.800 

23 

i     52 

53 

3     52 

28 

1.867 

58 

3-867 

29 

i     56 

59 

3     56 

29 

1-933 

59 

3-933 

30 

2         O 

60 

4       o 

30 

2.OOO 

60 

4.000 

196 


ASTRONOMY. 


To  convert  Sidereal  Time  into  Parts  of  the  Equator  in  Arc,  or  to 
convert  Time  into  Terrestrial  Longitude  in  Arc. 


HOURS. 

MINUTES. 

SECONDS. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

h. 

o 

m. 

0     , 

in. 

0     , 

s. 

,  „ 

s. 

,  „ 

i 

15 

i 

o  15 

31 

7  45 

I 

o  15 

31 

7  45 

2 

30 

2 

o  30 

32 

8  o 

2 

o  30 

32 

8  o 

3 

45 

3 

o  45 

33 

8  15 

3 

o  45 

33 

8  15 

4 

60 

4 

I   O 

34 

S  30 

4 

1   0 

34 

8  30 

5 

75 

5 

i  15 

35 

8  45 

5 

i  15 

35 

8  45 

6 

90 

6 

i  30 

36     9   O 

6 

i  30 

36    9  o 

7 

105 

7 

i  45 

37    9  15 

7 

i  45 

37 

9  15 

8 

1  20 

8 

2   0 

38    9  30 

S 

2   O 

3S 

9  30 

.    9 

135 

9 

2  15 

39 

9  45 

9 

2  15 

39 

9  45 

10 

150 

10 

2  30 

40   10  o 

10 

2  3O 

40 

10   0 

ii 

165 

ii 

2  45 

41 

10  15 

ii 

2  45 

4i 

10  15 

12 

180 

12 

3  o 

42 

10  30 

12 

3  o 

42 

10  30 

13 

195 

13 

3  15 

43 

10  45 

13 

3  15 

43 

10  45 

14 

210 

14 

3  30 

44 

II   O 

14 

3  30 

44 

II   O 

15 

225 

15 

3  45 

45 

ii  15 

15 

3  45 

45 

ii  15 

16 

240 

16 

4  o 

46 

ii  30 

16 

4  o 

46 

ii  30 

17 

255 

17 

4  15 

47 

ii  45 

17 

4  15 

47 

ii  45 

18 

270 

18   4  30 

48 

12   O 

18 

4  30 

48 

12   0 

19 

285 

19   4  45 

49 

12  15 

19 

4  45 

49 

12  15 

20 

3OO 

20   50 

50 

12  3O 

20 

5  o 

50 

12  30 

21 

315 

21   5  15 

5i 

12  45 

21 

5  15 

5i 

12  45 

22 

330 

22   |  5  30 

52 

13  o 

22 

5  30 

52 

13  o 

23 

345 

23   5  45 

53 

13  15 

23 

5  45 

53   13  15 

24 

360 

24 

6  o 

54 

13  30 

24 

6  o 

54 

13  30 

25 

6  15 

55 

13  45 

25 

6  15 

55 

13  45 

26 

6  30 

56 

14  o 

26 

6  30 

56 

14  o 

27 

6  45 

57 

14  15 

27 

6  45 

57 

14  15 

28 

7  o 

58 

14  30 

28 

7  o 

58   14  30 

29 

7  15' 

59 

14  45 

29 

7  15 

59 

14  45 

30   7  30 

60 

15  o 

30   7  30 

60 

15  o 

TIME    INTO    SPACE. 


I97 


To  convert  Sidereal  Time  into  Parts  of  the  Equator  in  Arc,  or  to 
convert  lime  into  terrestrial  Longitude  in  Arc — Continued. 


TENTHS  OF  SECONDS. 

Time.       Arc. 

Time.        Arc. 

Time. 

Arc. 

Time. 

Arc. 

s. 

,i 

s. 

s. 

" 

s. 

" 

o.oi           0.15 

0.31          4.65 

0.61 

9-15 

0.91 

13-65 

O.O2              0.30      |       0.32             4.80 

0.62 

9-30 

0.92 

13.80 

0.03 

0.45 

0.33         4-95 

0.63         9.45 

0-93 

13-95 

0.04 

0.60 

0.34         5-10 

o  .  64          9  .  60 

0.94 

.    14.10 

0.05 

•0.75 

0.35          5-25 

0.65          9.75 

0-95 

14-25 

O.O6 

0.90 

0.36         5-40 

0.66     ;     9.90 

0.96 

14.40 

0.07 

1.05 

0.37         5-55 

0.67     i   10.05 

0.97 

14-55 

0.08 

1.20 

0.38          5-70 

0.68        10.20 

0.98 

14.70 

O.Og 

1-35 

0.39         5-85          0.69        10.35 

0-99 

14.85 

O.  IO 

1.50 

0.40         6.00     ,     0.70     i  10.50 

i  .00 

1  5  .  oo 

O.  II 

1.65. 

0.41          6.15 

0.71     i  10.65 

_ 

0.12 

i.  So 

0.42       .6.30          0.72        10.80 

1 

0.13 

1-95 

0.43         6.45          0.73     ,   10.95 

c 

0.14 

2.10 

O.44                       6.6O              !            O.74                   II.    IO 

T3 

O 

0.15 

2.25 

0.45         r'-75 

0.75    ;  11.25 

s 

t-t-l 

Arc. 

0.16           2.40 

o  .  46          6  .  90 

0.76        11.40 

o 

CO' 

0.17           2.55 

0.47          7.05 

o.77     ;  H.55 

1 

o.iS 

2.70 

o  .  48         7  .  20 

0.78     i  11.70 

o 

0.19           2.85 

0.49         7-35 

0.79       11.85 

H 

o  .  20           3  .  oo 

0.50         7.50 

O.8O      ;    12.  OO 

s. 

0.21               3.15 

0.51          7.65 

0.81    ;  12.15 

O.OOI 

0.015 

0.22               3.30 

0.52         7.80 

0.82 

12.30 

O.OO2 

0.030 

0.23               3.45' 

0.53         7-95 

0.83 

12.45 

0.003 

0.045 

0.24 

3.60 

0.54         8.10 

0.84 

12.60 

O.OO4 

0.060 

0.25 

3-75 

0.55         ;         8.25 

0.85 

12.75 

0.005 

0.075 

O.26 

3-90 

O.56                 8.4O 

0.86 

12.90 

O.OO6 

o.ogo 

O.27 

4-05 

0.57         8.55 

0.87 

13-05 

O.OO7 

0.105 

0.28 

4.20 

6.58         8.70 

0.88 

13.20 

q.ooS 

O.  I2O 

0.29 

4-35 

0.59         8-85 

0.89 

^3-35 

o.oog 

0.135 

0.30 

4-50 

o  .  60         9  .  oo 

1 

0.90 

13-50 

O.OIO 

0.150 

198 


ASTRONOMY. 


To  convert  Right  Ascension  in  Arc  into  Mean  Time. 


DEGREES. 

AR. 

in  arc. 

Mean 

1 
time. 

AR. 

n  arc. 

Mean 

1 
time. 

AR. 

in  arc. 

Mean 

time. 

/, 

m. 

s. 

0 

/, 

in. 

s. 

0 

h. 

m. 

s. 

i 

o 

3 

59-345  | 

31 

2 

3 

39.686 

61 

4 

,  3 

20.027 

2      i 

o 

7 

58,689 

32 

2 

7 

39-030 

62 

4 

7 

I9.37I 

3 

o 

ii 

58.034  | 

33 

2 

ii 

38.375 

63 

4 

ii 

18.716 

4 

o 

T5 

57-379 

34 

2 

15 

37.720 

64 

4 

15 

18.061 

5 

•  o 

T9 

56.724 

35 

2 

19 

37.064 

65 

4 

19 

17-405 

6 

o 

23 

56.068 

36 

2 

23 

36.409 

66 

4 

23 

16.750 

7     : 

o 

27 

55-413 

37 

2 

27 

35-754 

67 

4 

27 

16.095 

8     : 

o 

3i 

54.758 

38' 

2 

3i 

35-099 

68 

4 

3i 

15-639 

9 

o 

35 

54.102 

39 

2 

35 

34-443  1 

4 

35 

14.784 

10 

o 

39 

53-447 

40 

2 

39 

33.788 

70 

4 

39 

14.129 

ii 

o 

43 

52.792 

41 

2 

43 

33-133  . 

7i 

4 

43 

13-474 

12 

o 

47 

52.136 

42 

2 

47 

32.477 

72 

4 

47 

12.818 

13 

o 

51 

51.481 

43 

2 

5i 

31.822 

73 

4 

51 

12.163 

14 

0 

55 

50.826 

44 

2 

55 

31.167 

74 

4 

55 

11.508 

15 

0 

59 

50.170 

45 

2 

59 

30.5H 

75 

4 

59 

10.852 

16 

I 

3 

49.515 

46 

3 

3 

29.856 

76 

5 

3 

10.197 

17 

I 

7 

48.860 

47 

3 

7 

29.201 

77 

5 

7 

9-542 

18 

I 

ii 

48.205 

48 

3 

ii 

28.545 

78 

5 

ii 

8.886 

19 

I 

15 

47-549 

49 

3 

15 

27.890 

79 

5 

15 

8.231 

20 

I 

19 

46.894 

50 

3 

T9 

27-235 

80 

5 

19 

7.576 

21 

I 

23 

46.239 

51 

3 

23 

26.580 

81 

5 

23 

6.920 

22 

I 

27 

45.583 

52 

3 

27 

25.924 

i     82 

5 

27 

6.265 

23 

I 

3i 

44.928 

53 

3 

3i 

25-269 

83 

5 

3i 

5.610 

24 

I 

35 

44-273 

54 

3 

35 

24.614 

84 

5 

35 

4-955 

25 

1 

39 

43.617 

55 

3 

39 

23.958 

85 

5 

39 

4.299 

26 

I 

43 

42.962 

56 

3 

43 

23-303 

86 

5 

43 

3.644 

27 

I 

47 

42.307 

57         3 

47 

22.648 

87 

5 

47 

2.989 

28 

I 

51 

41.652  !      58 

3 

5i 

21.992 

88 

5 

51 

2-333 

29 

I 

55 

40.996       .59 

3 

55 

21-337 

89 

5 

55 

1.678 

30 

I 

59 

40.341    |     60 

3 

59 

20.682 

90 

5 

59 

1.023 

RIGHT  ASCENSION  IN  ARC  INTO  MEAN  TIME. 


I99 


To  convert  Right  Ascension  in  Arc  into  Mean    Time — Continued. 


DEGREES. 


AR. 
in  arc. 

Mean  time. 

AR. 

in  arc. 

Mean  time. 

AR. 

in  arc. 

Mean  time. 

0 

Ji.     m.       s. 

o 

h.     m.       s. 

o 

//.     in.        s. 

9* 

6       3       0.367 

121 

8       2     40.708 

151 

IO      2      2I.O49 

92 

6       6     59-712 

122 

8       6     40.053 

152 

10     6     20.394 

93 

6     10     59.057 

123 

8     10     39.398 

153 

10  10     19.738 

94 

6     14     58.401 

124 

S     14     38.742 

154 

10  14     19.083 

95 

6    is   57-746  : 

125 

8     18     38.087 

155 

10  18     18.428 

96 

6     22     57.091 

126 

8      22      37-432 

156 

10   22       17.773 

97 

6     26     56.436 

127 

8     26     36.7/6 

157 

10  26     17.117 

98 

6     30     55.730 

128 

8     30     36.121 

158 

10  30     16.462 

99 

6     34     55.125        129 

8     34     35.466  !      159 

10  34     15.807 

IOO 

6     38     54-470        130 

8     38     34.810  ,      160 

10  38     15.151 

IOI 

6     42     53-SI4 

131 

S     42     34.155  j|     161 

10  42     14.496 

102 

6     46     53-159 

132 

8     46     33.500        162 

j 

10  46     13.841 

103 

6     50     52.504 

133 

8     50     32.845  !      163 

10  50     13.185 

104 

6     54     51-848 

134 

8     54     32.189        164 

10  54     12.530 

105 

6     58     5LI93 

135 

8     58     31-534 

165 

10  58     11.875 

I  O6 

7       2     50.538 

136 

9       2     30.879 

166 

II       2       11.220 

107 

7       6     49.883 

137 

9       6     30.223 

167 

ii     6     10.564 

108 

7     10    49.227 

138 

9     10     29.568 

168 

ii  10      9.909 

109 

7     14     48.572 

139 

9     14     28.913 

169 

II    14         9.254 

no 

7     18     47-9*7 

140 

9     18     28,257 

170 

ii  18       8.598 

III 

7     22     47.261 

141 

9      22      27.6O2 

171 

II    22         7.943 

112 

7     26     46.606 

142 

9     26     26.947 

172 

II    26         7.288 

H3 

7     30    45-  95i  _ 

143 

9     30     26.292 

173 

II    30         6.632 

114 

7     34     45-295 

144 

9     34     25.636 

J74 

ii  34       5-977 

H5 

7     38     44.640 

145 

9     38     24.981 

175 

ii  38       5-322 

116 

7     42-43.985 

I46 

9     42     24.326 

176 

n  42       4.666 

117 

7     46     43-329 

147 

9     46     23.670 

177 

IT  46       4.011 

118 
119 

7     50    42.674 
7     54     42.019 

I48 
I49 

9     50     23.015 
9     54     22.  -360 

178 
179 

ii  50       3.356 
ii   54       2.701 

1  20 

7     58     41.364  il     150 

9     58     21.704 

i  So 

ii   58       2.045 

200                                                         ASTRONOMY. 

To  convert  Right  Ascension  in  Arc  into  Mean  Time  —  Continued. 

MINUTES. 

SECONDS. 

AR. 

Mean 

AR. 

Mean 

AR. 

Mean 

AR. 

Mean 

in  arc. 

time. 

in  arc. 

time. 

in  arc. 

time. 

in  arc. 

time. 

' 

;;/.    s. 

' 

m.     s. 

» 

s. 

a 

s. 

i 

o     3.989 

3i 

2     3.661 

i 

0.066 

31 

2.061 

2 

o     7.978 

32 

2     7-650 

2 

0.133 

32 

2.128 

3 

o  11.969 

33 

2    II.64O 

3 

0.199 

33 

2.194 

4 

o  15.956 

34 

2    15.629 

4 

0.266 

34 

2.261 

5 

o  19.945 

35 

2    19.618 

5 

0.332 

35 

2.327 

6 

o  23.935 

36 

2    23.607 

6 

0-399 

36 

2-393 

7 

o  27.924 

37 

2    27.596 

7 

0.465 

37 

2.460 

8 

0   31-9*3 

38 

2    3L585 

8 

0.532 

38 

2.526 

9 

o  35-902 

39 

2    35-574 

9 

0.598 

39 

2-593 

10 

o  39.891 

40 

2    39.563 

10 

0.665 

40 

2.659 

j 

ii 

o  43.880 

41 

2    43-552 

ii 

0.731 

4i 

2.726 

12 

o  47.869 

42 

2    47-541 

12 

0.798 

42 

2.792 

13 

o  51.858 

43 

2    5L530 

13 

0.864 

43 

2.859 

14 

o  55.847 

44 

2    55.519 

14 

0.931 

44 

2.925 

15 

o  59-836 

45 

2    59.509 

15 

0.997 

45 

2.992 

16 

I    3.825 

46 

3     3.498 

16 

1.064 

46 

3.058 

i? 

I   7.814 

47 

3     7.487 

17 

1.130 

47 

3-I25 

18 

I  11.803 

48 

3  H.476 

18 

1.197 

48 

3-191 

19 

I  15.793 

49 

3  15-465 

19 

1.263 

49 

3.258 

20 

I  19.782 

50 

3  19-454 

20 

1-330 

50 

3.324 

21 

I  23.771 

5i 

3  23.443 

21 

1.396 

5i 

3.391 

22 

I  27.760 

52 

3  27.432 

22 

1.463 

52 

3-457 

23 

I  31.749 

53 

3  3L42I 

23 

1-529 

53 

3-524 

24 

I  35.738 

54 

3  35-410 

24 

1.596 

54 

3-59° 

25 

I  39.727 

55 

3  39-399 

25 

1.662 

55 

3-657 

26 

I  43.716 

56 

3  43.388 

-     26 

1.729 

56 

3.723 

27 

I  47.705  I 

57 

3  47.377 

27 

1-795 

57 

3.790 

23 

I  51.694 

58 

3  5L367 

28 

1.862 

58 

3.856 

29 

I  55.683 

59 

3  55.356 

29 

1.928 

59 

3-923 

30 

I  59.672 

60       3  59-345 

30 

1-995 

60 

3.989 

MEAN   TIME    INTO    RIGHT   ASCENSION    IN    ARC. 


201 


To  convert  Mean  Time  into  Right  Ascension  in  Arc. 


HOURS. 

MINUTES. 

I 

Mean 
time. 

AR.  in  arc. 

Mean 
time. 

AR.  in  arc. 

Mean 
time. 

AR.  in  arc. 

h. 

c 

m. 

0                 , 

m. 

0                , 

i 

15         2      27.85 

i 

O      15         2  .  46 

31 

7    46     16.39 

2 

30       4     52.69 

2 

o    30      4.93 

32 

8       i     18.85 

3 

45       7     23.54 

3 

o    45     30.39 

33 

8     16     21.31 

4 

60      9     51-39 

4 

i      o      9.86 

34 

8     31     23.78 

5 

75     12     19.24 

5 

I    15    12.32 

35 

8     46     26.24 

6 

90     14     47-oS 

6 

I    30   14.79. 

36 

9       i     28.71 

7 

105     17     14.93 

7 

i     45     17-25 

37 

9     16     31.17 

8 

120     19     42.78 

8 

2         O      19.71 

33 

9     3i     33-64 

9 

135  .  22     10.62 

9 

2       15       22.18 

39 

9     46     36.10 

10 

150     24     38.47 

10 

2      30      24.64 

40 

10       i     38.57 

ii 

165     27       6.32 

ii 

2     45     27.11 

41 

10     16     41.03 

12 

180     29     34.16 

12 

3       o     29.57 

42 

10     31     43-39 

13 

195       32         2.01 

13 

3     15     32.03 

43 

10    46    45.96 

14 

210     34     29.86 

14 

3     30     34-50 

44 

ii       i     48.42 

15 

225    36   57.70 

15 

3     45     SM6 

45 

ii     16     50.89 

16 

240     39     25.55 

16 

4       o     39.43 

46 

ii     3i     53-35 

17 

255     41     53.40 

17 

4     15     41.89 

47 

ii     46     55.81 

18 

270    44     21.24 

18 

4     30    44.35 

48 

12         I       58.38 

19 

285     46     49.09 

19 

4     45     46.82 

49 

12       17         0.74 

20 

300    49     16.94 

20 

5       o     49.28 

50 

12      32         3.21 

21 

315     51     44.78 

21 

5     15     5L75 

5i 

12     47       5-57 

22 

330     54     12.63 

22 

5     30     54-21 

52 

13         2         8.13 

23 

345     56     40.48 

23    . 

5     45     56.67 

53 

13     17     10.60 

24 

360     59       8.33 

24 

6      o     59.14 

54 

13     32     13.06 

25 

6     16       i.  60 

55 

13     47     15-53 

26 

6     31       4-°7 

56 

14       2     17.99 

27 

6     46       6.53 

57 

14     17     20.45 

28 

7       i       9  .  oo 

58 

14     32     22.92 

29 

7     16     11.46 

59 

M     47     25.38 

30 

7     3i     13.92 

60 

15       2     27.85 

ASTRONOMY. 


To  convert  Mean  Time  into  Right  Ascension  in  Arc — Continued. 


SECONDS. 

TENTHS  OF  SECONDS. 

Mean 
time. 

AR.  in  arc. 

Mean 
time. 

AR.  in  arc. 

Mean 
time. 

AR.  in 

arc. 

Mean 
time. 

AR.  in 
arc. 

s. 

;             // 

s. 

/         a 

s. 

,, 

s. 

;/ 

i 

o     15.04 

3i 

7     46.27 

O.OI 

0.15 

0.31 

4.66 

2 

o    30.08 

32 

S       1.31 

O.O2 

0.30 

0.32 

4.81 

3 

o    45.12 

33 

8     16.36 

O.O3 

0.45 

0.33 

4.96 

4 

i       o.  16 

34 

8     31.40 

O.O4 

0.60 

0-34 

5-12 

5 

i     15.21 

35 

8     46.44 

O.O5 

0-75 

0-35 

5-27 

6 

i     30.25 

36 

9       1.48 

0.06 

0.90 

0.36 

5.42 

7 

i     45.29 

37 

9     16.52 

O.O7 

'    1.05 

0-37 

5-57 

8 

2       0.33 

38 

9     31.56 

0.08 

1.20 

0.38 

5.72 

9 

2       15.37 

39 

9     46.60 

O.Og 

1-35 

0.39 

5.87 

10 

2      30.41 

40 

10       1.64. 

O.  IO 

1.50 

0.40 

6.  02 

ii 

2     45-45 

4i 

10     16.68 

O.II 

1.65 

0.41 

6.17 

12 

3       0.49 

42 

10     31.73 

O.  12 

i.  Si 

0.42 

6.32 

13 

3     15-53 

43 

10    46.77 

0.13 

1.96 

0-43 

6.47 

14 

3     30.53 

44 

ii       i.  Si 

0.14 

2.  II 

0.44 

6.62 

15 

3     45-  62 

45 

ii     16.85 

0.15 

2.26 

0.45 

6.77 

16 

4       0.66 

46 

ii     31.89 

o.  16 

2.41 

0.46 

6.92 

17 

4     15.70 

47 

ii     46.93 

0.17 

2.56 

0.47 

7-07 

18 

4     30.74 

48 

12         1.97 

0.18 

2.71 

0.48 

7.22 

19 

4     45.73 

49 

12       17.01 

0.19 

2.86 

0.49 

7-37 

20 

5       0.82 

50 

12      32.O5 

O.2O 

3.01 

0.50 

7-52 

21 

5     15-  86 

5i 

12      47.09 

0.21 

3-16 

0.51 

7.67 

22 

5     30.90 

52 

13         2.14 

O.22 

3-31 

0.52 

7.82 

23 

5     45-94 

53 

13       17.18 

0.23 

3.46 

0-53 

7-97 

24 

6       i.  oo 

54 

13      32.22 

O.24 

3-61 

0-54 

8..  1  2 

25 

6     16.03 

55 

13      47.26 

0.25 

3.76 

0-55 

8.27 

26 

6     31.07 

56 

14         2.30 

0.26 

3-91 

0.56 

8.43 

27 

6     46.11 

57 

14      17-34 

0.27 

4.06 

0-57 

8.58 

23 

7       i.i5 

58 

M      32.38 

0.28 

4.21 

0.58 

8.73 

29 

7     16.19 

59 

14      47.42 

O.29 

4.36 

0-59 

8.SS 

30 

7     31-23 

60 

15         2  .  46 

0.30 

4-51 

0.60 

9-°3 

MEAN    TIME    INTO    RIGHT    ASCENSION    IN    ARC. 


To  convert  Mean  Time  into  Ri?ht  Ascension  in  Arc — Continued. 


TENTHS    OF    SECONDS. 

THOUSANDTHS 
OF  SECONDS. 

Mean 
time. 

AR.  in 
arc. 

Mean 
time. 

AR.  in 
arc. 

Mean 
time. 

AR.in 

arc. 

Mean 
time. 

AR.  in 
arc. 

s. 

" 

s. 

" 

s. 

» 

s. 

» 

0.61 

9.18 

0.76 

11-43 

0.91 

13.69 

O.OOI 

0.02 

0.62 

9-33 

0.77 

11.58 

0.92 

13.84 

O.OO2 

0.03 

0.63 

9.48 

0.78 

11.74 

0-93 

13.99 

0.003 

0.05 

0.64 

9-63 

0.79 

11.89 

0.94 

14.14 

O.OO4 

0.06 

0.65 

9.78 

0.80 

12.04 

0.95 

14.29 

0.005 

o.oS 

0.66 

9-93 

o.Si 

12.  19 

0.96 

14.44 

O.OO6 

0.09 

0.67 

10.08 

0.82 

12.34 

0.97 

14.59 

0.007 

O.II 

0.68 

10.23 

0.83 

12.49 

0.98 

14.74 

O.OO8 

O.  12 

0.69 

10.38 

0.84 

12.64 

0.99 

14.89 

O.OOg 

0.14 

0.70 

10.53 

0.85 

12.79 

i  .00 

15-05 

O.OIO 

0.15 

0.71 

10.68 

o.S6 

12.94 

0.72 

10.83 

0.87 

13.09 

0.73 

10.98 

0.88 

13-24 

0.74 

11.13 

0.89 

13.39 

0-75 

11.28 

0.90 

13-54 

CONSTANT    LOGARITHMS. 


Logarithms. 

12  hours,  expressed  in  seconds  — 

43200 

4   6/3O8'37 

Complement  to  the  same                              — 

0000231  5 

c  05  j.^  16  ^ 

24  hours,  expressed  in  seconds  = 
Complement  to  the  same                              — 

86400. 
ooooi  157 

4.9365137 
c  o6'3J.S6'3 

360  degrees,  expressed  in  seconds  — 

1296000 

6  1126050 

To  convert  sidereal  time  into  mean  solar  time 

9  9988126 

204 


ASTROXOMY. 


FORM  FOR 
SURVEY  OF  DETERMINATION  or  TIME, 

DATE  AND  STATION. — 1843,  October  13. — Month  of  the  Big  Black  River, 
C  Sextant   No.   2197,  by    Troitghton    &°  Simms,    and 


INSTRUMENTS  ... 


Mean    Solar    Chronometer    No.     76,    by    Charles 


2  s 

*o  ^  i    ^     .            o   ^o 

£  £ 

o         ^ 

in   ,  '     o     *"     M 

u-» 

& 

|  *   55 

o   £J 

3    "o     c     £    *J* 

~  .2 

O     -*-»     "^j 

Names  of  stars. 

<U     "-3       ^ 

Hill 

u,   i   ! 

fj 

o  .^  *5 

?-*    *+J      <L>      i            «si       <U      "1> 
<j>      V-       £      O      c/5       |             £       ^       y 

«   c    S 

o  rt  ^ 

S^  8** 

^    °  * 

H 

0         /               // 

0           /           // 

h.  m.     s. 

/£.   ;;/.     J. 

91  43  40 

45  52  58.  8 

7  °5  47-  69 

6  57  02.4 

92  18  oo 

46  10  09.3 

7  07  28.67 

6  58  43.  2 

92  41  15 

46   21    47.3 

7  08  37.  15 

6  59  52.8 

a  Andromeda, 

93  04  05 

46  33  12.6 

7  °9  44.37 

7  oo  59.  6 

(cast.} 

93  45  20 
94  13  45 

46  53  50.  8 
47  08  03.  7 

7  ii  45.92 
7  13  °9-  73 

7  03  01.2 
7  04  25.  6 

94  40  50 

47  21  36.6 

7  14  29.64 

7  05  45- 

95  07  25 

47  34  54-  5 

7  15  48.  H 

7  °7  03-  6 

Mean  result  of  8  observations  on  a  Andromeda,  in  the  # 


T 

7     . 

95  20  05 

47  41  14.  7 

//.  ;;/,  s. 
8  55  32.36 

n.  in,  s. 
8  46  49.  2 

95  oo  oo 

47  3i  ii.  6 

8  56  32.06 

8  47  50.4 

94  3°  40 

47  16  31-2 

8  57  59.42 

8  49  1  6. 

a  Lyra 

94  12  20 

47  07  21. 

8  58  54- 

8  50  10.8 

(-vest  } 

93  53  45 

46  58  03.  i 

8  59  49.4 

8  51  06.9 

93  29  20 

46  45  50.  2 

9  01  02.  i 

8  52  19.4 

93  07  35 

46  34  57-3 

9  02  07. 

8  53  24.8 

92  46  50 

46  24  34.  5 

9  °3  °9- 

8  54  26. 

92  28  45 

46  15  3L7 

9  04  02.  96 

8  55  21.2 

Mean  result  of  nine  observations  on  the  star  a  Lyra,  in  the  west 

Mean  result  of  eight  observations  on  the  star  a  Andromeda,  in  the  east,  as  above . 

CHRONOMETER  ERROR.— Slow  of  mean  solar  time  at  8h  /.   m.t  by  a  mean 
of  these  results  from  east  and  west  stars 


TIME    BY    OBSERVED    ALTITUDES. 


RECORD  AND  COMPUTATION. 

by  Observed  Double  Altitudes  of  East  and  West  Stars. 

a  tributary  to  the  river  Saint  John,  Maine. 
artificial  horizon  of  Mercury. 

Younsr. 


Chronometer  (C.  Y. 
76)  slow  of  mean 
solar  time  by  each 
observation. 

Remarks. 

Index  error  of  sextant                                —  -f-  2'  40" 

O   08   4^    20 

Error  of  eccentricity  of  sextant                —  -1-  I   32 

8  45-47 
8  44-35 

8  44.    77 

Thermometer,  31°.  5  Fahrenheit. 
Barometer,  29.14  inches. 
Apparent  right  ascension  of  star    —   olloom2is.  72 

8  44  <;6 

Apparent  declination  of  star  —  •  28°  13'   59".  5  X. 

8  44.13 
8  44.  64 
8  44.  54 

Apparent  north  polar  distance  of  star  .  =  61    46    oo  .5   =  A 
Approximate  latitude  of  this  station  ..   —46    57    oo  N.  =  L 
Approximate  longitude  of  this  station  .  =    4h  37m47s 
Sidereal  time  of  mean  noon  at  station  .  =  13    26    20  .83 

oh  o8m  44s.  74 

//.  ;;/.     s. 
o  08  43.  16 
8  41.66 
8  43.  42 
8  43.  20 
8  42.  50 

8  4.2    7O 

Thermometer,  29°  Fahrenheit. 
Barometer,  29.  14  inches. 
Apparent  right  ascension  of  star  =  i8b  31"'  39".  16 
Apparent  declination  of  star  north  =  38°  38'   46".  5 
Apparent  north  polar  distance  of  star  .  =  5*    2I     !3  -5  —  A 
Index  error  of  sextant  ......       ......   —  ~  —  {-  2;  40" 

8  42  20 

Error  of  eccentricity  of  sextant                 —  -f-  I   32 

8  43.  oo 
8  41.  76 

oh  o8m42s.6 
o   08    44.7 

oho8m43s.6 

Observer,  Major  J.  D.  Graham. 
Computer,  Private  F.  Ilcrbst. 

206 


ASTRONOMY. 


Computation   of  the   Fifth   of  the  Preceding  Altitudes  of  a.  An 
dromeda,  (formula,  page  187.) 


Observed  double  altitude. 

Index  error,  sextant 

Eccentricity,  sextant 


=   93°  45'  20" 

=  +     02  40 

=  4-    01  32 


Double  altitude  corrected 
Altitude 


Refraction,  (thermom.,  31°.  5  ;barom..  29°.  i  )..  = 


93  49  32 
46  54  46 


56  .6 


True  altitude  of  *  =  A =      46    53   49  .4 


cos  = 
sin  = 

cos  L  sin  J  = 


9.8341894 
9.9449899 

9-779I793 


L=    46°57/ 

J  =    61    46  oo".5 

A  =    46    53  49.4 

2  w  =  155    36  49-9 
;;/  =    77    48  24.4 

(;;/  — A)  =    30    54  35.0 


sin-  J  p  = 


\p  —  25°o8/oi//.5 

p  in  arc  =  50  16  03  .o 

(page  192)  p  in  time  =  —    3  21  04  .20 

AR.  *  —  24  oo  21   .72 

Sidereal  time  of  observation. . .  =  AR.  ±  /  =  20  39  17. 52 
Sidereal  time,  mean  noon,   at  place,  (nauti 
cal  almanac) =  13  26  20  -83 


cos 

COS 

;//  sin 
m  sin 

cos  =. 
sin  = 

(m  -  A)  = 

(m  —  A) 

9.3247127 
9.7106984 

9.0354111 
19.2562318 
9.6281159 

cos  L 

sin  A 

sin 

4/  = 

Sidereal  interval  past  mean  noon  . 
Retardation  of    mean  on  sidereal  interval, 


(page  190) 


Mean    solar   interval   past   mean  noon,    or 

mean  time  p.  m.  of  observation 

Time  of  observation  by  chronometer 

"Chronometer  slow . .  


7    I2  56  -69 

=  —         01  10  .93 

7    II  45  -76 

7   03  01  .20 

8  44  .56 


OBSERVATIONS    FOR    THE    TIME.  207 


L. — To  Find  the  Time  by  Equal  Altitudes  of  the  Sitn. 

Correction  in  time,  to  be  applied  as  an  equation  to  the  mean 
of  the  times  of  observed  equal  altitudes  of  the  sun,  in  order  to 
obtain  the  time  of  its  meridional  passage  : 

T  T 

x  =  o  tan  D  -         — T-™  —  d  tan  L 


30  tan  7  J  T  "  30  sin  7^  T 

Make— 

T  T 

™  =  A  ;  -  .,  rr  ==  n 


30  sin  7£  T  "        '  30  tan  7^ 
x  =  =p  A  d  tan  L  +  B  d  tan  D 
Apparent  noon  =  J  (t  -f  /')  +  x 
/,  /'  =  the  times  of  observations  ; 

T  =  (t'  —  /)  =  the  interval  of  time  between  the  observations, 
expressed  in  hours  and  decimals  ; 

L  =  the  latitude  of  the  place  of  observation,  (myius  when 
south;) 

D  =  the  sun's  declination  at  apparent  noon  on  the  given  day, 
(mums  when  south  :) 

d  =  the  hourly  variation  in  the  declination  at  noon,  (minus 
when  the  sun  is  proceeding  toward  the  south ;)  and 

x  =  required  correction  in  seconds,  where  A  is  to  be  minus- 
where  the  time  of  noon  is  required  and  plus  where  the 
time  of  midnight  is  required,  i.  e.,  when  the  first  ob 
servation  is  made  in  the  afternoon  and  the  correspond 
ing  one  the  morning  following. 

Logarithmic  values  of  A  and  B  are  given  in  the  following 
tables. 


208                                                         ASTRONOMY. 

Equations  to  Equal  Altitudes. 

Interval. 

Log  A. 

LogB. 

Interval. 

Log  A. 

Log  B. 

h.     in. 

h.     in. 

2         O 

9.4109 

9-  3959 

3      o 

9.4172 

9-  3828 

2 

.4111 

•3955 

2 

.4174 

.3822 

4 

•4113 

•3952 

4 

.4177 

.3817 

6 

.4114 

.3948 

6 

.4179 

.3811 

8 

.4116 

•3944 

8 

.4182 

.3806 

10 

.4118 

•3941 

10 

.4184 

.  3800 

12 

.  4120 

•3937 

12 

.4187 

•3794 

14 

.  4121 

•3933 

H 

.4190 

.3789 

16 

.4123 

•  3929 

16 

•4193 

.3783 

18 

•  4125 

.3925 

18 

.4195 

•3777 

20 

.4127 

•3921 

20 

.4198 

•3771 

22 

.4129 

•391; 

22 

.4201 

.3765 

24 

•4I31 

.3913 

24 

.4204 

•3759 

26 

.4133 

.3909 

26 

.4207 

•3752 

28 

.4135 

.3905 

28 

.4209 

.3746 

3° 

.4137 

.3900 

30 

.  4212 

•3740 

32 

.4139   ' 

.3896 

32 

.4215 

•3733 

34 

.4141 

.3892 

34 

.4218 

.3727 

36 

.4144 

•  3887 

36 

.4221 

.3720 

38 

.4146 

.3882 

^o 

.4224 

.3713 

40 

.4148 

.3878 

40 

•4227 

.3707 

42 

.4150 

.3873 

42 

.4231 

.3700 

44 

•4152 

.3868 

44 

.4234 

•  3693 

46 

•4155  . 

•3863 

46 

.4237 

.3686 

48 

.4157 

.3859 

48 

.4240 

•  3679 

50 

.4159 

.3854 

50 

.4243 

.3672 

52 

.4162 

.3849 

52 

.4246 

.3665 

54 

.4164 

.3843 

54 

.4250 

.3657 

56 

.4167 

•3838 

56 

.4253 

•  3650 

2      58 

9.4169 

9-  3833 

3    58 

9.4256 

9-  3643 

x=  =p  A  o  tan  L  +  B  5  tan  D 

OBSERVATIONS    FOR   THE    TIME. 


209 


Equations  to  Equal  Altitudes — Continued. 


Interval. 

Log  A. 

LogB. 

Interval. 

Log  A. 

LogB. 

h,  m. 

h.  m. 

4   o 

9.  4260 

9-  3635 

5   o 

9-  4374 

9-  3369 

2 

.4263 

.3627 

2 

•4378 

•3358 

4 

.4266 

.  3620 

4 

.4383 

.3348 

6 

.4270 

•  3612 

6 

.4387 

•3337 

8 

•4273 

.3604 

8 

•  4391 

.3327 

10 

.4277 

•3596 

10 

.4396 

.33i6 

12 

.4280 

.3588 

12 

.4400 

•33°5 

H 

.4284 

.358o 

H 

•  4405 

•  3294 

16 

.4288 

•3572 

16 

.4409 

•3283 

18 

.4291 

•3564 

18 

.44H 

.3272 

20 

.4295 

•3555 

20 

.4418 

•  3261 

22 

.4299 

•3547 

22 

•  4423 

•  3249 

24 

.4302 

•3538 

24 

.4427 

•3238 

26 

.4306 

•3530 

26 

•  4432 

.3226 

28 

.4310 

•3521 

28 

•4437 

•  3214 

30 

•43  '4 

•3512  ' 

30 

.4441 

•  3203 

32 

•  43J7 

•3503 

32 

.  .4446 

•  3!9i 

34 

.4321 

•3494 

34 

.4451 

.3178 

36 

•4325 

.3485 

36 

•  4456 

.3166 

38 

•  4329 

•3476 

38 

.4460 

•3I54 

40 

•4333 

•3467 

40 

•  4465 

.  .3142 

42 

•4337 

•3457 

42 

.4470 

.3129 

44 

•  4341 

.3448 

44 

•4475 

.3116 

46 

•4345 

.3438 

46 

.4480 

•3103 

48 

•  4349 

•  3429 

48 

•  4485 

.3091 

5° 

•4353 

•3419 

5o 

.4490 

.3078 

52 

•4357 

.3409 

52 

•4494 

.3064 

•  54 

.4361 

•3399 

54 

.4500 

•3051 

56 

.4366 

•3389 

56 

•4505 

•3038 

4  58 

9-4370 

9-  3379 

5  58 

9.4510 

9-  3°24 

x  =  ^  A  d  tan  L  -f  B  d  tan  D 


210                                                      ASTRONOMY. 

Equations  to  Equal  Altitudes  —  Continued. 

Interval. 

Log  A. 

Log  B.           Interval. 

Log  A. 

LogB. 

h.     m. 

h.     m. 

6      o 

9.4515 

9.  3010 

7      o 

9.4685 

9-  2530 

2 

.4521 

.2996 

2 

.4691 

.  2511 

4 

.4526 

.2982 

4 

.4697 

.2492 

6 

•4531 

.2968 

6 

.4704 

•2473 

8 

.4536 

•  2954  , 

8 

.4710 

•2454 

10 

•  4542 

.2940 

10 

.4716 

•2434 

12 

•4547 

.2925 

12 

.4723 

.2415 

H 

•4552 

.2911 

H 

.4729 

•2395 

16 

.4558 

.2896 

16 

•4735 

•2375 

18 

•4563 

'.2881 

18 

.4742 

•2355 

20 

.4569 

.2866 

20 

.4748 

•2334 

22 

•  4574 

.2850 

22 

•4755 

•2313 

24 

.4580 

•2835 

24 

.4761 

.  2292 

26 

.4585 

.  2819 

26 

.4768 

.  2271 

28 

•4591 

.2804 

28 

•4774 

.  2250 

30 

•  4597 

.2788 

3° 

.4781 

.2228 

32 

.  4602. 

.2772 

32 

.4788 

.2206 

34 

-  .  4608 

.2756 

34 

•  4794 

.2184 

36 

.4614 

•2739 

36 

.4801 

.2162 

38 

.4620 

.2723 

38 

.4808 

.2140 

40 

.4625 

.2706 

40 

.4815 

.2117 

42 

.4631 

.2689 

42 

.4821 

.2094 

44 

.4637 

.2672 

44 

.4828 

.  2070 

46 

.4643 

•2655 

46 

.4835 

•  2047 

48 

.4649 

.2638 

48 

.4842 

.  2023 

5° 

.4655 

.  2620 

50 

.4849 

.1999 

52 

.4661 

.  2602 

52 

.4856 

.1974 

54 

.4667 

.2584 

•54 

.4863 

.1950 

56 

.4673 

.2566 

56 

.4870 

.1925 

6     58 

9.  4679 

9-  2548 

7    58 

9.  4877 

9.  1900 

a?  =  ^p  A  d  tan  L  +  B  d  tan  D 

OBSERVATIONS    FOR   THE    TIME.                                 211 

Equations  to  Equal  Altitudes  —  Continued. 

Interval. 

Log  A. 

Log  B. 

Interval. 

Log  A. 

Log  B. 

h.     m. 

h.     m. 

8      o 

9.  4884 

9.  1874 

9      ° 

9.5H5 

9-  0943 

2 

.  4892 

.1848 

2 

•5I23 

.  0906 

4 

.4899 

.  1822 

4 

•5132 

.0867 

6 

.4906 

.1796 

6 

.5140 

.0828 

8 

•49'3 

.1769 

8 

.5148 

.0789 

10 

.4921 

.1742 

10 

.5157 

.0749 

12 

.4928 

.1715 

12 

•5l65 

.0708 

14 

•4935 

.1687 

H 

.5174 

.0667 

16 

•  4943 

.1659 

16 

.5182 

.0625 

18 

•  4950 

.  1630 

18 

•5l9l 

•0583 

20 

.4958 

.  1602 

20 

.5199 

.0540 

22 

.4965 

•1573 

22 

.5208 

.0496 

24 

•4973 

•1543 

24 

•  5217 

.0452 

26 

.4980 

.1513 

26 

•5225 

.  0406 

28 

.4988 

.1483 

28 

.5234 

.0360 

3° 

.4996 

•1453 

30 

•5243 

.0314 

32 

•5003 

.1422 

32 

•5252 

.0266 

34 

.  5011 

.1390 

34 

.526! 

.0218 

36 

.5019 

•1359 

36 

.5269 

.0169 

38 

.5027 

.1327 

38 

.5278 

.    .0119 

40 

•5°35 

.1294 

40 

.5287 

.0069 

42 

.5042 

.  1261 

42 

.5296 

.  0017 

44 

•5050 

.  1228 

44 

.5305 

8.9965 

46 

.5058 

.1194 

46 

.5315 

.9911 

48 

.5066 

.1159 

48 

.5324 

.9857 

50 

•  5°74 

.1125 

50 

•5333 

.  9802 

52 

.5082 

.1089 

52 

•5342 

•  9745 

54 

.5091 

.1054 

54 

•5351 

.9688 

56 

•  5°99 

.  1017 

56 

.536i 

.9630 

8    58 

9.5107 

9.  0981 

9    58 

9.  5370 

8.9570 

x  —  if  A  d  tan  L  -+-  B  d  tan  D 

= 

2  I  2                      ASTRONOMY. 

Equations  to  Equal  Altitudes  —  Continued. 

Interval. 

Log  A. 

LogB. 

Interval. 

Log  A. 

Log  B. 

h.  tn. 

/i.  m.  < 

14   o 

9.  6841 

-9.0971 

15    0 

9-  7333 

-9.3162 

2 

.6856 

.1057 

2 

•7351 

•3225 

4 

.6872 

.  1141 

4 

•  7369 

.3287 

6 

.6887 

.  1224 

6 

.7386 

•335° 

8 

.6903 

•  13°6 

8 

.7404 

•  3411 

10 

.6919 

.1387 

10 

.7422 

•  3472 

12 

.6934 

.1468 

12 

.7440 

•3533 

H 

.6950 

.1547 

14 

.7458 

•3593 

16 

.6966 

.  1625 

16 

.7476 

.3653 

18 

.6982 

.1703 

18 

•  7494 

.3713 

20 

.6998 

.1779 

20 

•7512 

•  3772 

22 

.7014 

.1855 

22 

•7531 

•3831 

24 

.7030 

.1930 

24 

•7549 

.3889 

26 

.7047 

.2004 

26 

.7568 

•3947 

28 

.7063 

.2078 

28 

.7586 

.4005 

3° 

.7079 

.2150 

30 

.7605 

.4062 

32 

.7096 

.2222 

32 

.7624 

.4119 

34 

'  .7112 

.2293 

34 

.  7642  • 

.4175   % 

36 

.7129 

.2364 

36 

.7661 

.4232 

38 

.7146 

•2434 

38 

.7680 

.4288 

40 

.  7162 

.2503 

40 

.7699 

•4343 

42 

'  .7179 

.2571 

42 

.7718 

•  4399 

44 

.7196 

.2639 

44 

.7738 

•  4454 

46 

.7213 

.2706 

46 

•  7757 

•  45°9 

48 

.7230 

•2773 

48 

.7776 

•4563 

5° 

.7247 

.2839 

5° 

.7796 

.4617 

52 

•  7264 

.2905 

52 

.7815 

.4671 

54 

.  7281 

.2970 

54 

.7835 

.4725 

56 

.7299 

•3034 

56 

.7855 

•  4778 

•H  58 

9.7316 

-9.  3098 

15  58 

9-  7875 

-9.4831 

x  =  '  =p  A  8  tan  L  +  B  d  tan  D 

OBSERVATIONS    FOR    THE    TIME.                                 213 

Equations  to  Equal  Altitudes  —  Continued. 

Interval. 

Log  A. 

Log  B.           Interval. 

Log  A. 

Log  B. 

h,     m. 

h.     m> 

16      o 

9-  7^95 

-  9.  4884 

17      o 

9-  8539 

-9.6383 

2 

•  79^5 

•4937 

2 

.8562 

.6431 

4 

•  7935 

.4990 

4 

.8585                 .6478 

6 

•  7955 

.5042 

6 

.8608 

.6526 

8 

•7975 

•5°94 

8 

.8632 

.6573 

10 

.7996 

.5146 

10 

.8655 

.6621 

12 

.8016 

.5197 

12 

.8679 

.6668 

H 

.8037 

.5248 

H 

.8703 

•  6715 

16 

.8058 

•5300 

16 

.8727 

.6762 

18 

.8078 

•5351 

18 

•8751 

.6809 

20 

.8099 

.5401 

20 

.8775 

.6856 

22 

.8120 

•5452 

22 

.8799 

.6903 

24 

.  8141 

•5502 

24 

.8824 

.6949 

26 

.8162 

•5553 

26 

.  8848 

.6996 

28 

.8184 

•  5603 

28 

.8873 

•7043 

30 

.8205 

.5653 

30 

.8898 

.7089. 

32 

.8227 

.5702 

32 

.8923 

•  7136 

34 

.8248 

•5752 

34 

.8948 

.7182 

36 

.8270 

.5801 

36 

.8973 

.7228 

38 

.8292 

•5850 

38 

.8999 

.7275 

40 

.8314 

.5900 

40 

.9024 

.732i 

42 

.8336 

.5948 

42 

.9050 

.7367 

44 

.8358 

•  5997 

44 

•9°75 

.7413 

46 

.8380 

.6046 

46 

.9101 

•7459 

48 

.  8402 

.6094 

48 

.9127 

•75°5 

50 

.8425 

.6143 

50 

.9154 

.7552 

52 

.8447 

.6191 

52 

.9180 

•  7598 

54 

.8470 

.6239 

54 

.  9206 

.7644 

56 

.8493 

.6287 

56 

.9233 

.7690 

16    58 

9.8516 

-9-6335 

17    58 

9.  9260 

-9.7736 

x  =  =p  A  <5  tan  L  -f  B  <5  tan  D 

214                                                       ASTRONOMY. 

Equations  to  Equal  Altitudes  —  Continued. 

Interval. 

Log  A. 

Log  B. 

Interval. 

Log  A. 

LogB. 

h.     in. 

h.     m. 

18      o 

9.  9287 

-9.7782 

19      o 

0.0172 

-9.9167 

2 

•93H 

.7827 

2 

.0204 

.9213 

4 

•9341 

.7873 

4 

.0237 

.  9260 

6 

.9368 

.7919 

"6 

.0270 

•93°7 

8 

.9396 

•7965 

8 

.  0303 

•9355 

10 

.9424 

.8011 

10 

.0336 

.9402 

12 

•9451 

.8057 

12 

.0370 

•  9449 

H 

•9479 

.8103 

'4 

.0403 

•9497 

16 

.9508 

.8149 

16 

•0437 

•9544 

18 

•9536 

.8195 

18 

.0472 

•  9592 

20 

•  9564 

.8241 

20 

.0506 

.9640 

22 

•  9593 

.8287 

22 

.0541 

.9687 

24 

.9622 

.8333 

24 

.0576 

•9735 

26 

.9651 

.8379 

26 

.0611 

.9784 

28 

.9680 

.8425 

28 

.0646 

.9832 

3° 

'    .9709 

.8471 

30 

.0682 

.9880 

32 

•  9739 

•  8517 

32 

.0718 

.9929 

34 

.9769 

•  8563 

34 

•  0754 

-9.9977 

36 

.9798 

.8609 

36 

.0790 

—  o.  0026 

38 

.9829 

.8655 

38 

.0827 

.0075 

40 

.9859 

.8701 

40 

.0864 

.  0124 

42 

.9889 

.8748 

42 

.0901 

.0173 

44 

.9920 

.8794 

44 

•0939 

.  0223 

46 

.9951 

.8840 

46 

.0976 

.0272 

48 

9.  9982 

.  8887 

48 

.  1015 

.0322 

50 

o.  0013 

•8933 

5o 

•i°53 

.0372 

52 

.0044 

.8980 

52 

.  1092 

.0422 

54 

.  0076 

.9026 

54 

.1131 

•  0473 

56 

.0108 

.9073 

56 

.1170 

•  0523 

18    58 

o.  0140 

—9.9120 

19    58 

o.  1209 

—  0.0574 

x  =  =p  A  <5  tan  L  -f-  B  d  tan  D 

OBSERVATIONS  FOR  THE  TIME.             215 

Equations  to  Equal  Altitudes  —  Continued. 

Interval.' 

Log  A. 

LogB. 

Interval. 

Log  A. 

LogB. 

//.  m. 

h.  m. 

20   o 

o.  1249 

—o.  0625 

21    0 

o.  2623 

—  o.  2279 

2 

.  1290 

.0676 

2 

.2676 

•  2339 

4 

.133° 

.0727 

4 

.2729 

.  2401 

6 

.1371 

.0779 

6 

.2783 

.  2462 

8 

.  1412 

.  0830 

8 

.2838 

.2524 

10 

•  1454 

.0882 

10 

.2893 

.2587 

12 

.1496 

•  0935 

12 

.2949 

.  2650 

14 

.1538 

.0987 

14 

•  3°°5 

.2714 

16 

.1581 

.  1040 

16 

•  3°63 

.2778 

18 

.  1623 

.1093 

18 

.3120 

.2843 

20 

.1667 

.  1146 

20 

.3179 

.2909 

22 

.1711 

.  1  200 

22 

•3238 

•  2975 

24 

•1755 

•1253 

24 

.3298 

.3041 

26 

.1799 

.1308 

26 

•3359 

.3109 

28 

.1844 

.  1362 

28 

.3420 

.   .3177 

3° 

.1889 

.1417 

30 

.3482 

.3245 

32 

•1935 

.1472 

S2 

•3545 

.3315 

34 

.1981 

.1527 

34 

.3609 

.3385 

36 

.2028 

.  1582 

36 

.3674 

.3456 

38 

.2075 

.1638 

38 

•3739 

.3527 

40 

.  2122 

.1695 

40 

•3805 

•3599 

42 

.2170 

.1751 

42 

.3873 

•  3673 

44 

.2218 

.  i8oS 

44 

•  3941 

•3747 

46 

.2267 

.1866 

46 

.  4010 

.3822 

48 

.2316 

.1924 

48 

.4080 

.3897 

50 

.2366 

.1982 

5° 

•4151 

•  3974 

52 

.  2416 

.2040 

52 

.4223 

.4052 

54 

.  2467 

.2099 

54 

.4297 

.4130 

56 

.2518 

•  2159 

56 

•  4371 

.4210 

20   58 

o.  2570 

—o.  2219 

21   58 

0.4446 

—0.4291 

x  =  =f  A  3  tan  L  +  B  d  tan  D 

2l6 


ASTRONOMY. 


Equations  to  Equal  Altitudes — Continued. 


Interval. 

Log  A. 

LogB. 

Interval. 

Log  A. 

LogB. 

7z.  m. 

h.  m. 

22    0 

o-  4523 

—0.4372 

23   o 

o.  7689 

—o.  7652 

2 

.4601 

•4455 

2 

.7842 

.7807 

4 

.4680 

•  4540 

4 

.8000 

.7967 

6 

.4761 

.4625 

6 

.8163 

•8i33 

8 

.4^42 

.4711 

8 

•8333 

•8305 

10 

.4926 

•4799 

10 

.8508 

•  8483 

12 

.  5010 

.4889 

12 

.8691 

.8667 

14 

.5097 

.4980 

14 

.8882 

.8860 

16 

.5184 

.5072 

16 

.9080 

.9060 

18 

•5274 

•5165 

18 

.9288 

.9270 

20 

•5365 

.5261 

20 

.9506 

•  9489 

22 

•5458 

•5358 

22 

•9734 

.9719 

24 

•5553 

•5457 

24 

o.  9975 

—  o.  9961 

26 

.5649 

•5557 

26 

i.  0228 

—  I.  0216 

28, 

.5748 

.5660 

28 

.0497 

.0487 

30 

.5848 

.5764 

30 

.0783 

.0774 

32 

•5951 

•  5871 

32 

.  1089 

.  1081 

34 

.  6056 

•5979 

34 

.  1416 

.  1409 

36 

.  6164 

.  6090 

36 

.1770 

.1764 

38 

.6273 

.6204 

38 

•  2154 

.2149 

40 

.6386 

.6319 

40 

•?573 

.2569 

42 

.6501 

.6438 

42 

•3°37 

•3033 

44 

.6619 

.6559 

44 

•3554 

•3552 

46 

.6740 

.6684 

46 

.4140 

.4138 

48 

.6865 

.6811 

48 

•  4815 

.4814 

50 

•  6993 

.6942 

50 

•  56l3 

.5612 

52 

•  7I24 

.7076 

52 

.6588 

.6587 

54 

.7259 

.7214 

54 

.7844 

.7843 

56 

.7398 

•7355 

56 

i.  9610 

—  1.9610 

22   58 

o.  7541 

—o.  7501 

.   23  58 

2.  2627 

—  2.  2627 

x  =  =p  A  3  tan  L  -f  B  d  tan  D 


EQUATION    OF    EQUAL    ALTITUDES.  2 17 

Computation  of  the  Equation  of  Equal  Altitudes  to  correct  the  Chro 
nometer  for  Noon,  August  9,  1844,  by  the  First  of  the  following 
Equal  Altitudes  of  the  Sun's  Limbs. 

x  =  (—  A  3  tan  L)  +  (B  d  tan  D) 

T  =    611  33m         log  A  =  —  9.4605             log  B  =  9.2764 

d   =  43".63            log  d    =  —  1.6397              log  d  =  —  1.6397 

L  =  45°  48'   log  tan  L  =        0.0121       log  tan  D  =  9.4493 


ist  term  i2s-95  =  1-1123  —  28.32=  —  0.3654 

2d  term  —  2  .32 

I 

-|-  10  .63  =  equation  of  equal  altitudes. 


Computation  of  the  First  Two  of  the  following  Pairs  of  Equal  Alti 
tudes  of  the  Sun's  Limbs. 

ist  pair.  ad  pair. 

A.  M.  /  =  i11  28m  233.o         ih  29™  52S.8 

P.M.  =         /'  =  8    03     16.5         8    01     46.5 


=  9    31     39-5        9    31     39  -3 


_±_  =  4    45     49  .75      4    45     49  .65 
Equat'n  of  equal  alts.  =         x   =          +  I0  -63  10  .63 


Time  by  chron.  of  appt.  noon  =  4    46     oo  .38      4    46     oo  .28 
Correct  mean  time  at  apparent 

noon  (Naut.  Aim.)  =  o    05     09  .09      o    05     09  .09 


Chron.  fast  of  mean  time  at  app't 

noon,  August  9,  1844  =  4    40     51  .29      4    40     51  .19 


2l8 


ASTRONOMY. 


SURVEY  OF  ... 


.   DETERMINATION  OF  THE  TIME, 
Chronometer 


DATE  AND  STATION. — 1844,  August  9 — American  Camp,  near  Tasche 


INSTRUMENTS 


^  Sextant  No.  2197,  by  Trough  ton  &* 
Mean  Solar  Chronometer,  No.  2440, 


Times,  by  chronometer,  of  ob 

Observed   double 
altitudes  of  the 
sun's  upper 
and      lower 

served  equal  altitudes. 

2"  —  t  =  the 
elapsed  time, 
=T. 

Equation 
of  equal 
altitudes 

August  gth. 

limbs. 

=  X. 

A.  M.  =  / 

P.  M.  =  f 

Upper  Limb. 

//.  m.  s. 

h.  ;.Y.  s. 

h.    m. 

s. 

78°  50'   oo" 

I  28  23 

8  03  16.  5  i 

6    33 

+10.63 

79      9    3° 

I  29  52.  8 

8  01  46.  5  ) 

Lower  Limb. 

83°  10'    oo" 

i  45  01 

7  46  40.5] 

83    40    oo 

i  46  34.  5 

7  45  °6-  2  j> 

5    59i 

+10.24 

84    oo    oo 

i  47  38 

7  44  04     J 

Upper  Limb. 

85°  36'    oo" 

i  49  23 

7  42  18      j 

5    48 

+  10.  I 

87      02      10 

i  53  55-5 

7  37  46.  2  5 

CHRONOMETER  ERROR. — Fastoi  mean  solar  time  at  apparent  noon  of  Au 
gust  9,  1844,  by  a  mean  of  7  pairs  of  equal  altitudes  of  the  sun 


TIME    BY    OBSERVED    EQUAL   ALTITUDES. 


219 


by  Observed  Equal  Altitudes  of  the  Sun's  Limbs,  to   Correct  the 
at  Noon. 

reau's  house,  on  the  highland  boundary  between  Maine  and  Canada. 

Simms,  and  Artificial  Horizon  of  Mercury. 
by  Parkinson  6°  Frodsham. 


Chron  o  m  e  t  e  r 
No.  2440  fast 
of  mean  time 
at  apparent 
noon  by  each 
pair  of  equal 
altitudes. 


h.  m.  s. 

4  40  5i-29 
4  40  51-  *9 


f4  40  5T-9 
<J4  40  51-5 
U  40  52.15 


4  40  5T-5i 
4  40  51.  86 


Remarks. 


Index  error  of  sextant 

Error  of  eccentricity  of  sextant 

Thermometer  (a.  m.)  70°  Fahr.  barom-. 

Thermometer  (p.  m.)  69°  Fahr.  barom. .. 

Sun's  appar't  declination  at  appar't  noon(D)=  15°  43'  12"  N. 

Hourly  variation  of  sun's  declination. .  .(r5)  =  43//-63 

Equation  of  time  at  apparent  noon -+-  5 '"  09*. 09 

Latitude  of  station  (approximate) -f-  45°  48'  =  (L. ) 


4  40  51.6 


Observer,  Major  J.  D.  Graham. 
Computer,  Do. 


220 


ASTRONOMY. 


Sun's  Parallax  in  Altitude. 


Sun's 

Sun's  horizontal  parallax. 

Sun's 

Sun's  horizontal  parallax. 

altitude. 

altitude. 

8".  7 

8".8 

8".9 

9".o 

8".  7 

8".8 

8".9 

9".o 

0 

H 

n 

„ 

n 

0 

n 

u 

n 

n 

0 

8.  70 

8.80 

8.90 

9.00 

45 

6.  iS 

6.22 

6.  29 

6.36 

5 

8.67 

8.77 

8.87 

8.97 

50 

5-59 

5.66- 

5.72 

5-79 

10 

8.57 

8.67 

8.76 

8.86 

55 

4.99 

5-05 

5-12 

5.16 

15 

8.40 

8.50 

8.60 

8.70 

60 

4-35 

4.40 

4-45 

4.5° 

20 

8.18 

8.27 

8.36 

8.46 

65 

3.68 

3-72 

3.76 

3.80 

25 

7.88 

7.98 

8.06 

8.16 

70 

2.98 

3.01 

3.04 

3.08 

30 

7-53 

7.62 

7.70 

7-79 

75 

2.25 

2.28 

2.31 

2-33 

35 

7-13 

7.21 

7.29 

7.37 

80 

1.51 

i-53 

1-55 

1.56 

40 

6.66 

6-74 

6.82 

6.  90 

85 

o.  76 

o.77 

0-77 

0.78 

45 

6-15 

6.22 

6.29 

6.36 

90 

o.  oo 

o.  oo 

o.  oo 

o.  oo 

Parallax  in  altitude  •==.  horizontal  parallax  X  cosine  of  altitude. 
Decimals  of  an  Hour. 


Minutes. 

Seconds. 

m. 

deem. 

m. 

deem. 

m. 

deem. 

s. 

deem. 

S. 

deem. 

s. 

deem. 

i 

.  01667 

21 

•  35°oo 

41 

•  68333 

i 

.  00028 

21 

.  00583 

41 

.01139 

2 

•  °3333 

22 

.36667 

42 

.  70000 

2 

.  00056 

22 

.  00611 

42 

.  01167 

3 

.  05000 

23 

.38333 

43 

.  71667 

3 

.  00083 

23 

.  00639 

43 

.01194 

4 

.  06667 

24 

.  40000 

44 

•  73333 

4 

.00111 

24 

.  00667 

44 

.01222 

5 

•  08333 

25 

.41667 

45 

.  75000 

5 

.  00139 

25 

.  00694 

45 

.  OI25O 

6 

.  IOOOO 

26 

•  43333 

46 

.  76667 

6 

.  00167 

26 

.  00722 

46  .01278 

.   7 

.  11667 

27 

.  45000 

47 

•  78333 

7 

.00194 

27 

.  00750 

47'  .01306 

8 

•  J3333 

28 

.46667 

48 

.  80000 

8 

.  OO222 

28 

.  00778 

48:  .01333 

9 

.  15000 

29 

.48333 

49 

.81667 

9 

.  00250 

29 

.  00806 

49 

.01361 

10 

.  16667 

3° 

.  50000 

50 

.83333 

10 

.  00278 

3" 

.  00833 

50 

.01389 

ii 

•  18333 

3i 

.51667 

5i 

.  85000 

ii 

.  00306 

31 

.  00861 

5i 

.01417 

12 

.  20000 

32 

.  53333 

S2 

.  86667 

12 

•  00333 

32 

.  00889 

52 

.01444 

13 

.21667 

33 

.  55000 

53 

.88333 

13 

.00361 

33 

.00917 

'53 

.01472 

H 

•  23333 

34 

.56667 

54  .  90000 

14 

.00389 

34 

.00944 

54 

.  01500 

15 

.  25OOO 

35 

.  5  33o 

55 

.91667 

15 

.00417 

35 

.  00972 

55 

.01528 

16 

.  26667 

36  i  .  60000 

56  .93333 

16 

.  00444 

36 

.  OIOOO 

S6 

.01556 

17 

•  28333 

37  .61667 

57  .95000 

17 

.  00472 

37 

.  01028 

57 

.01583 

-i  8 
19 

.  3OOOO 
.31667 

38  j  .  63333 
39  .  65000 

58 
59 

.96667 
•  98333 

18 
19 

.00500 
.  00528 

38 
39 

.  01056 

.  01083 

58 
59 

.  Ol6ll 
.  01639 

20 

•  33333 

40  .  66667 

60 

I.  00000 

20 

.  00556 

40 

.  OIIII 

60 

.01667 

« 

TIME    BY    TRANSITS.  221 


LI. — The  Transit  Instrument, 

Knowing  the  apparent  right  ascension  of  a  star,  to  compute 
the  corrections  to  its  observed  transit  on  account  of  the  three 
principal  errors  of  the  transit  instrument — in  azimuth,  in  the 
inclination  of  the  axis,  and  in  collimation — in  order  to  obtain 
the  correct  clock  error : 

E  -  T  +  a  *Hk::J9  +  b  ^  (L  ~  D)  +  ~  -  AR. 

cos  D  cos  D  cos  D 

where — 

E  denotes  the  error  of  the  clock,  minus  when  slow ; 
T,  the  observed  time  of  transit ; 
L,  the  latitude  of  the  place ; 

D,  the  declination  of  the  star,  plus  when  north,  and  minus 
when  south,  for  the  upper  culminations,  and  vice  versa  for 
the  lower  culminations ; 

a,  the  deviation  in  the   telescope    in   azimuth,  plus  when 
(pointing  to   the   south)   the  vertical  which  it  describes 
falls  to  the  east,  and  minus  when  it  falls  to  the  west,  and 
vice  versa  when  pointing  to  the  north ; 

b,  the  bias  or  inclination  of  the  axis  of  the  telescope,  plus 
when  the  west  end  of  the  axis  is  too  high ; 

c,  the  error  in  collimation,  plus  when  the  circle  described 
by  the  line  of  collimation  of  the  telescope  falls   to  the 
east,  and  minus  when  it  falls  to  the  west,  for  upper  cul 
minations,  and  vice  versa  for  lower  culminations ;  and 

AR.,  the  right  ascension  of  the  star.^ 

When  the  clock  marks  mean  solar  time,  the  mean  time  of 
transit  of  the  object  over  the  meridian  must  be  substituted 
for  AR. 


222  ASTRONOMY. 


LI.  —  The  Transit  Instrument  —  Continued. 

i.  To  determine  the  value  (in  time)  of  the  co  -efficients  0, 
in  the  preceding  formula  : 

For  inclination  of  the  axis  of  the  telescope  : 


where  — 

w'  and  e1  denote  respectively  the  values  of  w  and  e  after 
reversing  the  level; 

d,  the  value  of  each  division  of  the  level  in  seconds  of  space; 
w,  the  inclination  of  the  level  to  the  west;  and 

e,  the  inclination  of  the  level  to  the  east. 

for  cottimation  : 

c  =  -J  (/'  -  t)  cos  D  +  J  (b1  -  b)  cos  (L  -  D) 

where  — 

f  and  b'  denote  respectively  the  values  of  /  and  b,  after 
reversing  the  instrument  ,• 

D,  the  declination  of  a  circumpolar  star;  and 
/,  the  time  of  the  transit  of  the  circumpolar  star  deduced 
from  an  observation  at  a  given  side  wire  of  the  instrument. 

For  the  deviation  -in  azimuth  : 

By  observations  of  a  circumpolar  star  : 

a  =  I2h  ~  (T/  ~  T)  +  b  CQS  (L  -  D)  -  V  cos  (L  +  D)  +  2  c 
2  cos  L  tan  D  2  cos  L  sin  D 

where  — 

T7  and  V  denote  respectively  the  values  of  T  and  b  at  the 
lower  culmination. 


TIME    BY    TRANSITS.  223 


LI.  —  The  Iransit  Instrument  —  Continued. 
Deviation  in  azimuth  by  transits  of  a  high  and  low  star  : 


where  — 

T',  AR/,  and  Dv  denote  respectively  the  values  of  T,  AR., 

and  D  of  the  second  star  observed. 
Or— 


- 
: 


cos  L  (tan  D  —  tan  D') 

If  one  of  the  stars  is  observed  at  its  lower  culmination,  use 
180°  —  D'  and  i2h  -f  AR/  for  its  declination  and  right  as 
cension. 

Or  make — 

sin  (L  -  D)  for 


cos  D 
and — 

sinJL_-  DO  for 
cos  D'  - 

then — 

_  (AR/  —  AR.)  —  (T  —  T) 
n1  —  n 

n  is  negative  for  a  star  north  of  the  zenith. 

2.  To  find  the  equatorial  interval  of  each  wire  from  the  central 
wire,  observe  the  transit  of  a  star  of  any  decimation  D ;  then — 

Equatorial  interval  =  observed  interval  x  cos  D. 

3.  When  the  intervals  on  each  side  of  the  central. wire  are 
equal,  the  mean  of  the  times  of  transit  over  each  wire  will  denote 
the  transit  over  the  middle  wire.     But  should  they  not  be  equal, 
a  correction  must  be  applied  to  obtain  a  correct  mean. 

Call  I.  II;  IV.  V,  the  equatorial  intervals  of  each  wire  from 
the  central  wire,  the  instrument  having  say  5  wires ;  then — 


fl  _L_  il\  (IV  4-  V) 

Reduction  to  middle  wire  =  ^ — — 

5  cos  D 


224                                                         ASTRONOMY. 

FORM  FOR  RECORD  AN 
SURVEY  OF 

Transits  of  Stars  .                   7c*i//i 

D  COMPUTATION. 
STATION. 

Inch  transit  No  ; 

Hardy  No.  50. 

Sidereal  Chronometer 

Illuminated  end  of  axis,  west. 

Date  (1847).      •- 

October  6th. 

October  6th. 

October  6th. 

Observer 

T.J.L. 

T.J.L. 

T.J.L. 

•      Object  

TT  Capricorni. 

14  Capricorni. 

a  Cygni. 

Level  

E.32.2  W.33.o 
£.32.2  W.33.o 

£.32.7  W.32.5 
E-32-5  W.33-3 

£.32.7  W.32.5 
E.33.o  W.32.5 

Value  of  i  division  of 
scale  —  7"  '.  Z 

Wires  I 

h.     m,       s. 

20       17      33.0 
17      53-5 

18    12.  7 

18    32.7 
20    18    52.  5 

h.     m.       s. 
20     29     43.7 
30     02.7 

30      22.  0 
•     30      4L7 

20    31     oo.  7 

h.     m.       s. 
20    35    oo.  o 
35    26.0 
35    52.0 
36    18.7 
20    36    45-5 

II 
III 
IV 
V 

Sum 

184,4 

1  10.  8 

142.2 

Mean 

20     18     12.88 
.07 

20      30      22.  1  6 

—      .07 

20    35    52.44 

—           .  10 

Reduc'n  to  middle  wire 

Transit  on  instrument. 
£    ^  ^  for  collimation  . 
"l:  .§   )>  for  level  .. 

12.  8l 
+            .10 

+       -17 

22.  09 

4-       .04 

+           -IS 

52.39 

—            .  12 
—            .01 

tj  •£  }  fordev'n  in  az'h 

Transit  by  chronom'r. 

20     1  8     13.08 

20      30      22.31 

20    35     52.21 

AR.  of  star  

20      1  8      36.66 

20      30      45.  89 

20    36     15.  So 

Error  of  chronometer  . 

23.58 

23.58 

23-59 

Chronometer    
at 

-    slow    of    time 
-  p.m.,  October  6th,  1847. 

TRANSIT    INSTRUMENT. 


225 


Computation  of  the  Corrections  a  and  b,  in  the  Preceding  Transits. 
Declination  of  TT  Capricorni  =  18°  42  S. 
14  Capricorni  =  15°  29  S. 
a  Cygni  =  44°  44  N. 
Latitude  of  Station  =  L  =  43°i3/ 
Level  Correction  of  x  Capricorni. 


=       43°  '3'  E.  32.2 


D  =  -  1  8°  42' 


-  D)  =       6.0  55' 

cos  <L  ~  D)  =  o.5< 
cos  D 


32.2 


33 
33 


64.4  66 

66  —  64.4  =  i.(> 

b  —  ?~5.  X  1.6  =  o".2o 
60 

7  cos  (L  —  D) 

Level  correction  =  o - — — — '-  =  os.2o  x  o.c;o  =  os.io 

cos  D 

Deviation  in  Azimuth. 
(AR/  -  AR.)  -  (T'  -  T) 
n'  -  n 

T'  and  T  being  the  times  of  transit  corrected  for  level  and  col- 

limation. 
Combining  -  Capricorni  and  a.  Cygni, 

h.     in.        s. 

AR/  =  20  36  15.80 
AR.    =  20  18  36.66 


h.    m.         s. 

T'  =  20  35  52.22 
T   =  20  18  12.91 


I7  39-31 


39-31 


(AR/  -  AR.)  -  (T  -  T)  =  -  0.17 

(sinL  -  DO       sin  (-1°  31') 


cos  D' 


k   cos44°44r 


sin  (L  —  D)  _  sin6i°  55' 
cosD         =  cos  1  8°  42' 


:  + 


a  = 


=       oM8 


—  0.03  —  0.93        0.96 

Combining  14  Capricorni  and  a  Cygni,  a  =  +  os.i9 
Correction  for  deviation  in  azimuth  of  JT  Capricorni, 


226 


ASTRONOMY. 


Numerical  Values  of  Factors 

>in  (L  —  D)     cos  (L  —  D) 

cos  D                cos  D       ' 

For  deviation. 

Star's  declination  —  ±  D 

For  level. 

Star's  Z.-D. 

Star's  Z.-D.    . 

=  (L-D) 

o° 

10° 

20° 

25° 

30° 

35° 

40° 

=  (L—  D) 

0 

0 

I 

.02 

.  02 

.02 

.02 

.02 

.02 

.02 

89 

2 

.04 

.04 

.04 

.04 

.04 

.04 

•05 

88 

4 

.07 

.07 

.07 

.08 

.08 

.08 

.09 

86 

6 

.  ii 

.  II 

.  II 

.  II 

.  12 

•13 

-H 

84 

8 

.14 

.14 

•15 

•15 

.16 

•17 

.18 

82 

10 

•17 

.18 

•19 

.19 

.20 

.21 

•23 

80 

12 

.21 

.21 

.22 

•23 

.24 

•25 

.27 

78 

14 

.24 

•25 

.26 

.27 

.28 

.29 

•32 

76 

16 

.28 

.28 

.29 

•30 

.32 

•34 

.36 

74 

18 

•31 

•31 

•33 

•34 

.36 

.38 

.40 

72 

20 

•34 

•35 

.36 

.38 

.40 

.42 

•45 

70 

22 

•37 

.38 

.40 

.42 

.44 

.46 

•  49 

68 

24 

.41 

.41 

•43 

•45 

•  47 

•  49 

•53 

66 

26 

.44 

•45 

•47 

.49 

•5i 

•54 

•57 

64 

28 

•47 

•48 

•50 

•52 

•54 

•57 

.61 

62 

30 

•So 

•51 

•53 

•55 

.58 

.61 

•65 

60 

32 

•53 

•54 

.56 

.58 

.61 

.65 

•  69 

58 

34 

.56 

•57 

•59 

.61 

.65 

-69 

•73 

56 

36 

•59 

.60 

•63 

.65 

.68 

.72 

•77 

54 

38 

.62 

•63 

.66 

.68 

•71 

•75 

.80 

52 

40 

.64 

.65 

.68 

•71 

•74 

.-78 

.84 

50 

45 

•71 

.72 

•75 

.78 

.82 

.86 

.92 

45 

So 

-77 

.78 

.82 

.84 

.89 

•93 

I.OO 

40 

55 

.82 

•83 

.87 

.90 

•95 

.98 

1.07 

•35 

60 

.87 

.88 

.92 

•95 

I.OO 

i.  06 

1.  13 

30 

65 

.91 

.92 

.96 

I.OO 

1.05 

I.  10 

1.  18 

25 

70 

.94 

•95 

I.OO 

1.04 

1.09 

1.14 

1.23 

20 

75 

•97 

.98 

1.03 

1.07 

I.  12 

1.17 

1.26 

15 

80 

.98 

I.OO 

1.05 

1.09 

I.I4 

1.20 

1.29 

IO 

89 

I.OO 

i.  02 

i.  06 

I.  10 

I-I5 

1.22 

1-31 

I 

For  colli-  ? 

/- 

I 

mation  ) 

I.OO 

i.  02 

I.  OO 

I.  IO 

I.  15 

I.  22 

I-3I 

—  cosD 

TRANSIT    INSTRUMENT. 


227 


1        for  1 

"acitttating  the  Reduction  of  Transit-  Observations. 

cos  D'  J    J 

For  deviation. 

Star's  declination  =  ±  D 

For  level. 

Star's  Z.-D. 

Star's  Z.-D. 

=  (L-D) 

45° 

50° 

55° 

60° 

65° 

70° 

75° 

=  <L-D) 

I 

.02 

•03 

•03 

•03 

.04 

•05 

.07 

89 

2 

•05 

•05 

.06 

.07 

.08 

.  IO 

.13 

88 

4 

.  10 

.  ii 

.  12 

.14 

•17 

.20 

.27 

86 

6 

•15 

.16 

.18 

.21 

•25 

.31 

.40 

84 

8 

.20 

.  22 

.24 

.28 

•33 

.41 

•54 

82 

10 

•25 

.27 

•30 

•35 

.41 

•51 

•  67 

80 

12 

.29 

•32 

•36 

•42 

.49 

.61 

.80 

78 

H 

•34 

•38 

.42 

•  48 

•57 

.71 

•  94 

76 

16 

•39 

•43 

.48 

•55 

.65 

.81 

i.  06 

74 

18 

•  44 

•  48 

•54 

.62 

•73 

.90 

1.19 

72 

20 

•  48 

•53 

.60 

.68 

.81 

I.OO 

1.32 

70 

2Z 

•53 

•58 

•65 

•75 

•  89 

1.09 

1-45 

68 

24 

•58 

-63 

•  7i 

.81 

.96 

1.  19 

i-57 

66 

26 

.62 

.68 

•  76 

.88 

1.04 

1.28 

1.69 

64 

28 

.66 

•73 

.82 

•  94 

i.  ii 

1.37 

1.81 

62 

30 

•  7i 

•78 

•  87 

I.OO 

i.  18 

1.46 

1.93 

60 

32 

•75 

.82 

.92 

i.  06 

1.25 

i-55 

2.05 

58 

34 

•79 

•87 

•97 

I.  12 

1.32 

1.65 

2.16 

56 

36 

•83 

.91 

1.03 

1.18 

1-39 

1.74 

2.27 

54 

38 

•  87 

.96 

1.07 

1.23 

1.46 

i.  80 

2.38 

52 

40 

.91 

I.OO 

I.  12 

1.29 

1.52 

1.88 

2.48 

5° 

45 

I.OO 

I.  10 

1.23 

1.41 

1.67 

2.07 

2-73 

45 

50 

i.  08 

1.19 

1-34 

r-53 

1.81 

2.24 

2.96 

40 

55 

1.16 

1.27 

1-43 

1.64 

1.94 

2.40 

3.16 

35 

60 

1.22 

1.35 

I-51 

1-73 

2.05 

2-53 

3-35 

30 

65 

1.28 

1.41 

1.58 

1.81 

2.14 

2.65 

3-50 

25 

70 

I-33 

1.46 

i.  64 

1.88 

2.22 

2-75 

3-63 

20 

75 

1-37 

1.50 

1.68 

1-93 

2.  29 

2.82 

3-67 

15 

80 

1-39 

r-53 

1.72 

i.  97 

2-33 

2.88 

3-8i 

10 

89 

1.41 

1.56 

i*74 

2,00 

2-37 

2.92 

3.86 

I 

Forcolli-  > 

T 

mation  )  *  " 

1.41 

1.56 

1,74 

2.00 

2-37 

2.92 

3-86 

~~cos  D 

228  ASTRONOMY. 


LI  I. — Reduction  of  Transits  by  Least  Squares. 
Let— 

E  be  the  error  of  chronometer  at'  an  assumed  time  T  ; 
A?  h)  ^3>  &c.,  the  observed  times  of  transit   (corrected   for 
rate  and  level  error)  of  stars  having  the  right  ascensions 
AR.b  AR.2,  AR.3,  &c.; 

a  and  c,  the  errors  of  azimuth  and  collimation ;  and 
A!,  A2,  A3,  £c.,  Ci,  C2,  C3,  &c.,  the  factors  of  azimuth  and 

of  collimation  for  the  several  stars ; 
then — 

/!  -j-  E  +  A!  a  +  Ci  c  =  ARt 

/2  +  E  +  A2  a  +  C2  c  —  AR2 
/3  +  E  +  A3  a  +  C3  c  =  AR3 

&c.,  &c. 
Let— 

E  =  E  +  e 

where  e  is  the  unknown  correction  to  an  assumed  chronometer 
error  E; 

and  let,  also, 

ARi  -  /i  =  e, 

AR2  —  /2  =  e-2 

:  AR3  -  /3  =  ^3 

&c.,  &c. 
then— 

£  +  e  +  A!  ^  -f  Ci  c  =  cl 

E  +  s  +  A2  «  +  C2  c  =  e9 
E  +  e  +  A3  «  +  C3  c  =  ^ 

&c.,  &c. 
Let  now — 

ei  —  E  =  tii 

^2   —    E  —   7/2 
€3   —    E   =.   7/3 

&c.,  &c. 
then — 

e  +  A!  #  +  Ci  £  =  7/! 

£    -j-    A2  «    +    C2  r    =    772 

e  4-  A3  a  +  C3  ^  ;=  7/3 

&c.,  &c. 
From  which  form  the  normal  equations — 

S£+ZAa+ZCc=n 

^Ae  +  ^A2^  4-  -TAC^  =  A;/  (i) 

JCe  +  £AC0 +  £€,<:  =  <:« 
from  which  e,  ^,  and  c  can  be  obtained. 


TRANSIT    INSTRUMENT. 


229 


LII. — Reduction  of  Transits,  <5rV. — Continued. 

If  the  errors  of  collimation  are  known,  and  the  times  /i,  /2>  ?3> 
&c.,  corrected  for  it,  the  azimuthal  deviation  and  correction  to 
assumed  chronometer-error  may  be  deduced  from  the  equations — 


2  s  +    I 


a  =      n 


Equations  (i)  cannot  be  advantageously  employed  unless  the 
nstrument  be  reversed. 

Example  of  the  Computation  of  Equations  (i). 

* 

Latitude,  36°  38'  N.— April  u,  1852— Assumed  time,  T  =  nh  sidereal— Chronometer 
losing  is .83  daily—Assume  E  =  +  3h  14'"  30^.0. 


Illum'n  star. 

• 

n 

A 

C 

A» 

C« 

A2 

AC 

C2 

f  a  Urs.  Maj  

h  .  in  .    s. 
3  *4  29-77 
29.91 
29.87 
28.50 
30.56 
30.19 
30.06 

-0.23 
-0.09 
-0.13 
-1.50 
+  0.56 
+  0.19 
+  0.06 

-0-95 
+  0.27 
+  0.80 
+  4.05 
—0.55 
+  0.46 
+  0.46 

+2.17 

+  1.07 

+  1.04 

+  4-39 
-1.72 

—  I.  01 

—  i.  02 

+  0.22 
—  O.O2 
-0.10 
—  6.07 
-0.31 

+0.09 
+  0.03 

-6.16 

—0.50 

—  O.  IO 

—0.14 

-6.58 

—0.96 
—0.19 
—  0.06 

+  0.90 
0.07 
0.64 
16.40 
0.30 

O.2I 
O.2I 

—  2.06 
+  0.29 
+  0.83 
-17.78 
+  0.95 
—  0.46 
—  °-47 

+  4.71 
1.14 

i.  08 
19.27 
2.96 
i.  02 
1.04 

+31.22 

(S  Hydra.     ... 

r  y  Cephei  sub.  polo 
U  Urs.  Maj  
'  1  4052  B.  A.  C  
[4072  B.  A.  C  

—  1.14 

+  4-54 

+  4.92 

-8.53 

+  18.73 

+  16.86 

Normal  Equations. 

7  £  +  4.54  a  +  4.92  c  =  —  1.14 
4.54  e  +  J8.73  a  +  16.86  c  =  —  6.16 
4.92  e  -|-  16.86  a  -\-  31.22  c  =  —  8.53 

from  which — 

£    =    -j-    O8.C>9 

a  —  —  os.i8 
c  •=.  —  o8. 19 

hence, 

Azimuthal  deviation  of  the  instrument =  os.i8  W.  of  S. 

Error  of  collimation  of  mean  of  wires,  illumi 
nation  east =  os.  1 9  W. 

Error  of  chronometer,  (slow) =  3Tl  14™  30". 06 


230  ASTRONOMY. 


.— Tables  of  Refraction. 

Table  I  gives  the  refraction  when  the  barometer  stands  at  30 
inches  and  the  Fahrenheit  thermometer  at  50°. 

Table  II,  to  be  used  when  greater  accuracy  is  desired,  gives 
the  correction  of  the  mean  refraction  depending  upon  the  ob 
served  height  of  the  barometer  and  thermometer. 

In  column  A  of  this  table,  the  refraction  is  regarded  as  a  func 
tion  of  the  apparent  zenith-distance  Z.  The  adopted  form  of 
this  function  is — 

r  =  a,3A  ?x  tan  Z 

in  which  a  varies  slowly  with  the  zenith-distance,  and  its  loga 
rithm  is  therefore  readily  taken  from  the  table  with  the  argument 
Z.  The  exponents  A  and  A  differ  sensibly  from  unity  only  for 
great  zenith-distances,  and  also  vary  slowly;  their  values  are 
therefore  readily  found  from  the  table. 

The  factor  y?  depends  upon  the  barometer.  The  actual  press 
ure  indicated  by  the  barometer  depends  not  only  upon  the 
height  of  the  column,  but  also  upon  its  temperature.  It  is 
therefore  put  under  the  form — 

/?  =  B  T 

and  log  B  and  log  T  are  given  in  the  supplementary  tables  with 
the  arguments  "Height  of  the  barometer"  and  "Height  of  the 
attached  thermometer,"  respectively ;  so  that — 

log  ?  =  log  B  +  log  T 

Finally,  log  Y  is  given  directly  in  the  supplementary  table  with 
the  argument  "  External  thermometer."  This  thermometer  should 
be  so  exposed  as  to  indicate  truly  the  temperature  of  the  atmos 
phere  at  the  place  of  observation. 


REFRACTION. 


23I 


L 1 1 1 . — Tables  of  Refraction — Continued. 

Example. — Given  the  apparent  zenith  distance,  Z  =  78°3o/o//; 
barometer,  29.770  inches;  attached  thermometer,  —  o°.4  F. ; 
external  thermometer,  —  2°.o  F. 

From  table  II  for  78°3o';  log  a  =  1.74981 
A  =  1.0032  ;  A  =  1.0328 


and  from  the  tables  for  barometer  and  thermometer — 

log  f  =  +  0.04545 


log  B  =  +  0.00253 
log  T  =  +  0.00127 


log  ft  =  -f-  0.00380 
Hence  the  refraction  is  computed  as  follows : 

log  a  =        1.74981 

A  log  /?  =  log  /5A  =  +  0.00381 

A  log  Y  —  l°g  VK  =  +  0.04694 

log  tan  Z  =        0.69154 

r  =  5'  io".53  =  3io".53;  log  r  =        2.49210 
The  true  zenith-distance  is,  therefore, 

78°  30'  +  s'io".S3  =  78°35'io".53 

TABLE  I. — Mean  Refraction. 
Barometer,  30  inches — Fahrenheit  thermometer,  50°. 


Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction. 

0             / 

/       // 

0             / 

/       // 

0              / 

/       // 

5    30 

9      7-0 

6    30 

7    53-9 

0        0 

36    29 

35 

9      o.  i 

35 

7    48.7 

I         0 

24    54 

40 

8    53-4 

40 

7    43-5 

2        0 

18    26 

45 

8    46.8 

45 

7    38.4 

*^        O 

14    25 

50 

8    40.4 

5° 

7    33-5 

4      o 

ii     44 

55 

8    34-2 

55 

7    28.6 

5      ° 

9    52-° 

6      o 

8    28.0 

7      o 

7    23.8 

5 

9    44.0 

5 

8      22.1 

5 

7    19.2 

10 

9    36.2 

10 

8      !6.2 

10 

7    14-6 

15 

9    28.6 

15 

8     10.5 

15 

7    10.  i 

20 

9      21.2 

20 

8      4.8 

20 

7      5-7 

25 

9     14.0 

25 

7    59-3 

25 

7      1-4 

232 


ASTRONOMY. 


TABLE  I. — Mean  Refraction — Continued. 
Barometer,  30  inches — Fahrenheit  thermometer,  50°. 


Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction. 

7    30 

6    57-1 

10    30 

5      4-6 

13     30 

3    58.1 

35 

6    53-0 

35 

5      2.3 

35 

3    56.6 

40 

6    48.9 

40 

5      o.  o 

40 

3    55-2 

45 

6    44.9 

45 

4    57-8 

45 

3    53.7 

50 

6    41.  o 

50 

4    55-6 

5° 

3    52.3 

55 

6    37-1 

55 

4    53.4 

55 

3    5°-9 

8      o 

6    33-3 

II         0 

4    51.2 

14      o 

3    49-5 

5 

6    29.6 

5 

4    49.i 

5 

3    48.  i 

IO 

6    25.9 

IO 

4    47.o 

10 

3    46.8 

15 

6    22.3 

15 

4    44.9 

15 

3    45-5 

20 

6    18.8 

20 

4    42.9 

20 

3    44-2 

25 

6     15.3 

25 

4-     4-O«  9 

25 

3    42.9 

30 

6     ii.  9 

30 

4    38.9 

30 

3    41.6 

35 

6      8.5 

35 

4    36.9 

35 

3    40.3 

40 

6      5-2 

40 

4    35-0 

40 

3    39-0 

45 

6        2.  0 

45 

4    33-1 

45 

3    37-7 

5° 

5    58.8 

5° 

4    31.2 

5° 

3    36.5 

55 

5    55-7 

55 

4    29.4 

55 

3    35-3 

Q        O 

5    52.6 

12        0 

4    27.5 

15      o 

3    34.i 

5 

5    49-6 

5 

4    25.7 

5 

3    32.9 

10 

5    46.6 

IO 

4    23.9 

10 

3    31-7 

15 

5    43-6 

15 

•    4      22.2 

15 

3    30.5 

20 

5    40.7 

20 

4    20.4 

20 

3    29.4 

25 

5    37-9 

25 

4     18.7 

25 

3    28.2 

30 

5    35.i 

3° 

4     17.0 

30 

3    27.1 

35 

5'  32.4 

35 

4     15-3 

35 

3    25.9 

40 

5    29.6 

40 

•4     13.6 

40 

3    24.8 

45 

C        27,  O 

45 

4     12.0 

45 

3    23.7 

50 

5    24.3 

5° 

4     10.4 

5° 

3    22.6 

55 

5    21.7 

55 

4      8.8 

55 

3    21.5. 

10        0 

5    19.2 

13      o 

4      7-2 

16      o 

3    20.5 

5  * 

5    16.7 

5 

4      5-6 

5 

3     !9-4 

10 

5    H.2 

10 

4      4.  i 

IO 

3     18.4 

15 

5    11.7 

15 

4      2.6 

15 

3     17.3 

20 

5      9-3 

20 

4       i.o 

20 

3     16.3 

25 

5      6-9 

25 

3    59-6 

25 

3    15-2 

TABLES    OF    REFRACTION. 


233 


TABLE  I. — Mean  Refraction — Continued, 
Barometer,  30  inches— Fahrenheit  thermometer,  50' 


Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean  re 
fraction. 

01                                     III 

, 

/       // 

16    30           3     14.2 

20      o          2    38.  8 

26      o 

I     58.9 

35          3     13-2 

10               2      37.4 

10 

i     58.  I 

40 

3     I2-2 

20      j         2      36.  0 

20 

i    57-2 

45 

30          2    34.  6 

30 

i    56.4 

50 

3     IO-3 

40 

2     33-3 

40 

i    55-5 

55 

3      9-3 

50               2      32.0 

50 

i    54-7 

17      o 

3      8.3     i;     21      o     !      2    30.7 

27      o 

i    53-9 

5 

3      7-3 

10 

2     29.4 

IO 

i    53-  1 

10 

3      6.4 

2O 

2      2-8.1 

20 

i    52.3 

15 

3      5-5 

30 

2      26.9 

30 

i    51-5 

20 

3      4-6 

40 

2      25.7 

40 

i    5o.7 

25 

3      3-7 

50 

2      24.5 

50 

i    50.0 

30 

3      2.8 

22        0               2      23.3 

28      o 

i    49.2 

35 

3      1-9 

10 

2      22.  I 

10 

i    48.4 

4° 

3       l-° 

2O 

2      20.9 

20 

i    47-7 

45 

3      o.  i 

30 

2      19.8 

30 

i    46.9 

50          2    59.2 

40 

2      18.7 

40 

i    46.2 

55          2    58.3 

50 

2      17-5 

50 

i    45-5 

18      o 

2      57.5 

23      o 

2       16.4 

29      o 

i    44.8 

5          2    56.6 

10 

2      15.4 

20 

i     43-4 

IO 

2    55-8    j            20 

2      14.3 

40 

i     42.0 

15 

2      54-9                       30 

2      13.3 

30      o 

i     40.  6 

20 

2    54-1 

40 

2      12.  2 

20 

i     39.3 

25 

2    53-2 

5° 

2      II.  2 

40 

i     38-0 

3° 

2      52.4      j|      24         0 

2      10.2 

31      o 

i    36.7 

35 

2      51-6      |                 10 

2        9.  2 

2O 

i    35-5 

40 

2      50.8 

20 

2         8.2 

40 

i    34-2 

45 

2      50.0 

30 

2         7.2 

32      o 

i    33-0 

2      49.2 

40 

2        6.2 

20 

i    31-8 

55 

2      48.4 

50 

2         5-3 

40 

i    30.7 

19      o 

2      47-7             25         0 

2        4-4 

33      o 

i    29.5 

10 

2      46.  I                        10 

2        3.4 

20 

i    28.4 

20 

2      44.  6                       20 

2        2.5 

40 

i    27.3 

3° 

2      43-1                        3° 

2         1.6 

34      o 

i    26.2 

40 

2      41.6                       40               2        0.  7                       20 

i    25.  i 

50 

2     40.  2                  50     I       i     59.8      ;             40 

i    24.  i 

234 


ASTRONOMY. 


TABLE  I. — Mean  Refraction — Continued. 
Barometer,  30  inches — Fahrenheit  thermometer,  50°. 


Apparent 
altitude. 

Mean    re 
fraction. 

Apparent 
altitude. 

Mean    re 
fraction*. 

Apparent 
altitude. 

Mean    re 
fraction. 

°         / 

/       // 

o         , 

/       // 

o          / 

/       // 

35      o 

I     23.  i 

47      o 

o    54-3 

59      o 

o    35-0 

20 

I      22.0 

20 

o    53-7 

20 

°    34-5 

40 

I      21.0 

40 

o    53-1 

40 

o    34.1 

36      o 

I      2O.  I 

48      o 

o    52.5 

60      o 

o    33-6 

20 

I       19.  I 

20 

o    51.9 

20 

o    33-2 

40 

I       18.2 

40 

o    51.2 

40 

o    32.7 

37      o 

I       17.2 

49      o 

o    50.  6 

61      o 

o    32-3 

20 

I       I6.3 

20 

o    50.  o     i 

62      o 

o    31.0 

40 

I       15.4 

.40 

o    49.4 

63      o 

o    29!; 

38      o 

I       14.5 

50      o 

o    48.  9 

64      o 

o    28.4 

2O 

I       13-6 

20 

o    48.3 

65      o 

o    27.2 

40 

i     12.7 

40 

o    47.8 

66      o 

o    25.9 

39      o 

I     11.9 

5i      ° 

o    47.2 

67      o 

o    24.7 

20 

I       II.  0 

20 

o    46.6 

68      o 

o     23.6 

40 

I       IO.  2 

40 

o    46.1 

69      o 

o    22.4 

40      o 

I         9.4 

52      o 

o    45-5 

„    70      o 

O      21.  2 

20 

i      8.6 

20 

o    45.0 

71      o 

0      20.  I 

40 

i       7.8 

40 

o    44.4 

72      o 

o     18.9 

41      o 

i       7.0 

53      o 

0    43-9 

73      o 

o     17.8 

20 

I         6.2 

20 

o    43-4 

74      o 

o    16.  7 

'  40 

i      5-4 

40 

o    42.8 

75      o 

o     15.6 

42      o 

i      4.7 

54      o 

o    42.3 

!   76    o 

o     14.5 

2O 

i      3-9 

20 

o    41.8 

77      o 

o    13-5 

40 

i      3.2 

40 

o    41.3 

78      o 

o     12.4 

43      ° 

i      2.4 

55      o 

o    40.8 

79      o 

o     11.3 

20 

i       1.7 

20 

o    40.3 

80      o 

o     10.3 

40 

I         I.O 

40 

o    39.8 

81      o 

o      9.2 

44      o 

i      0.3 

56      o 

o    39-3 

!    82      o 

0        8.2 

20 

o    59.6 

20 

o    38.8 

83      o 

o      7.2 

40 

o    58.9 

4° 

o    38.3 

84      o 

o      6.1 

45      o 

o    58.2 

57      o 

o    37.8 

85      o 

o      5.1 

20 

o    57.6 

20 

o    37-3 

86      o 

o      4.  i 

40 

o    56.9 

40 

o     36.9 

87      o 

o       3.1 

46      o 

o    56.2 

58      o 

o    36.4 

88      o 

0        2.0 

20 

o    55-6 

20 

o    35-9 

89      o 

O         I.O 

40 

0    55-° 

40 

o    35-5 

90      o 

0        0.0 

REFRACTION. 


235 


TABLE  II. — BesseVs  Refraction- Table. 


i 

1 

Arg.app.  zen.-dist. 

8 

Arg.  app.  zen.-dist. 

« 

T 

'S 

Q 

N 

Log  a. 

A 

» 

'S 

4) 

N 

Log  a. 

A 

» 

0     / 

O    O 
10    0 

20   o 

1.76156 

1.76154  . 
.76149 

77   o 

IO 
20 

1-75229 

*•  75205   _^ 

1-75180  y 

1.0026 
i  .0026 
1.0027 

1.0252 
1.0258 
1.0264 

30   o 
35   ° 

1.76139  9 
1.76130  * 

3° 
40 

1.75155  J 

i-75I29  28 

1.0027 
i  .0028 

1.0272 
I.02§I 

40   o 

5° 

i.75ioi  2g 

1.0029 

1.0290 

45   o 
46   o 

48   o 

1.76104 
1.76100  * 
1.76096  J 
1.76092  4 

I.OOlS 

I  .OOI9 
1.0019 
I  .0020 

78   o 

IO 
20 
30 

1.75072 
i  •  75°43  ,o 
1-75013  „ 
i.7498i  H 

1.0030 

1.0030 
1.0031 
1.0032 

1.0299 
1.0308 
1.0318 
1.0328 

1.76087  5 

I.OO2I 

40 

1  •  74947  :£. 

1.0033 

1.0338 

50   o 

1.76082  * 

1.0023 

50 

1.749"  ^g 

1.0034 

1-0347 

• 

52   o 
53   o 
54   o 
55   o 
56   o 

1.76077  6 
1.76071  6 
i  .  76065 
1.76058  I 
1.76050  g 
1.76042  g 

1.0025 
I.OO26 
1.0027 
I  .OO29 
I.OO3I 
1.0034 

79   ° 

IO 
20 
30 
40 

5° 

1.74876 
1.74839  *' 
1-74799  72 
1-74757  J. 
I'747I4  . 
1.74670  ™ 

1.0035 
1.0036 
1.0037 
1.0038 
1.0039 
1.0040 

1.0357 
1.0367 
1-0377 
1.0387 
1.0398 
1.0409 

57   o 
58   o 
59   o 
60   o 
61   o 

1.76033  I0 
1.76023  „ 
1.76012 
1.76001 
1.75988  *3 

1.0037 

I  .0040 
1.0043 
1  .0046 
1.0049 

80   o 

IO 
20 

30 
40 

1-74623  -0 
1-74573  * 
I-7452I  | 
1.74468  53 
I-744I2  £o 

1.0041 

1.0042 
1.0043 
1.0045 
i  .0046 

I.O420 
I.043I 
1.0442 

1.0454 
i  .  0466 

62   o 

1-75973  jg 

1.0054 

50 

1-74352  64 

1.0047 

1.0479 

63   o 
64   o 
65   o 
66   o 
67   o 
68   o 

1-75957  l8 
1-75939  20 

I.759I9  22 

1.75897  26 
I.75871  2Q 
1-75842  33 

1.0058 
1.0063 
1.0068 
1.0075 
1.0083 
1.0092 

81   o 

10 

20 

30 
40 
50 

1.74288  g 
1.74223  6§ 
I.74I55  72 
1-74083  '6 
1.74007  7 
1.73928  g 

1.0049 
1.0050 
1.0052 
1.0054 
1.0056 
1.0058 

1.0493 
1.0508 
1.0523 
1.0540 
1-0559 
1-0579 

69   o 
70   o 
71   o 
72   o 
73   ° 

1-75809  g 
I-7577I  ;L 
1.75726  ^ 

I  .OIOI 
I.OIII 

1.0124 
1.0139 
1.0156 

82   o 

10 

20 

30 
40 

1.73845  gS 

I'73W  94 
1-73663  94 

1.73564  J0^ 
1-73459  H2 

i.  0060 
1.0062 
1.0065 
1.0067 
1.0070 

I.  0600 
i  .  0622 
1.0646 
1.0671 
1.0697 

74   o 

1-75543  H 

1.0175 

5° 

1-73347  Il8 

1.0073 

1.0725 

75   o 

10 

20 

i  -75457  l6 
I-7544I  l6 
1.75425  I7 

1.0197 

I.O2OO 
I  .O2O4 
1.  0208 

83   o 

IO 

20 

3° 

1-73229  I2. 
I-73I05   4 

1.72974  ^ 

1.72832  3* 

1.0075 
1.0078 
i.  0081 
1.0084 

1.0754 
1.0784 
1.0815 
1.0846 

40 

1.75391  { 

I.O2I2 

40 

1.72681  *g 

i.  0088 

1.0879 

50 

1.75373  Ig 

I.  O2l6 

50 

I.725I9  I73 

1.0092 

1.0914 

76   o 

IO 

1-75355  IQ 
1.75336  y 

I.O22O 
1.0225 

84   o 

10 

1.72346  l86 
1.72160 

1.0096 

I.OIOO 

1.0951 
1.0992 

20 

f-  20 

1.0230 

20 

1.71961   " 

1.0105 

I  .  1036 

3° 

1.75295  2: 

1.0235 

30 

I.7I749  227 

I.  OIIO 

1.1082 

40 
50 

I-75274  22 

1.0241 
I  .  0246 

40 
50 

I-7I522    * 
I.7I279  J« 

1.0115 

I.OI2I 

1.1130 
1.1178 

77   o 

1.75229 

1.0026 

I.O252 

85   o 

I.7IO2O 

I.OI27 

1.1229 

236 


ASTRONOMY. 


TABLE 

1  1  .  —  BesseVs  Refraction-  Table. 

Factor  depending  upon 
the  barometer. 

Eng.  ins 

Log  B. 

Factor  depending  upon  the  external 
thermometer. 

F. 

Log  y. 

F. 

Logy. 

27-5 

—  0.03191 

27.6 

27.7 

0.02876 

° 

0 

27.8 
27.9 
.  28.0 

0.02720 
0.02564 
0.02409 

—  20 
18 

+  0.06279 
o.  06181 
o  .  06083 

+3i 

3.6 

37 

+  0.01185 
0.01098 

O.OIOII 

28.2 
28.3 
28.4 
28.5 
28.6 
28.7 
28  8 

0.02254 
0.02099 
0.01946 
0.01793 
0.01640 
0.01488 
0.01336 

3 

15 
J4 
13 

12 
II 

0.05985 

0.05887 
0.05790 
0.05693 
0.05596 
0.05500 
0.05403 

38 
39 
40 

41 
42 

43 

44 

0.00924 
0.00837 
0.00750 
0.00664 
0.00578 
0.00492 
0.00406 

28.9 
29.0 
29.1 
29.2 

29-3 

29.4 

29-5 
29.6 

29-7 
29.8 

O.OIO35 

0.00885 
0.00735 
0.00586 
0.00438 
0.00290 
—0.00142 
+  0.00005 
0.00151 
0.00297 

10 

9 
8 

0 

5 
4 
3 

2 

—  I 
0 

0.05307 
0.05211 
0.05115 
0.05020 
0.04924 
0.04829 
0.04734 

o  .  04640 

0-04545 
0.04451 
0-04357 

45 
46 

3 

49 
50 
51 
52 
53 
54 
55 

0.00320 

0.00234 

0.00149 
+  0.00064 

—  0.00021 
O.OOIO6 
O.OOigi 
0.00275 

o  .  00360 

0.00444 

0.00528 

29.9 
30.0 
30.1 
3°-2 
30-3 
30-4 
3°-5 
30.6 
3°-7 
30.8 
3°-9 
31-0 

0.00443 
0.00588 
0.00732 
0.00876 

O.OIO2O 
0.01163 
0.01306 
0.01448 
0.01589 
0.01731 
0.01871 
+  O.O2OI2 

+  1 
2 

3 

4 

I 

9 

10 

ii 

•12 

0.04263 
0.04169 
0.04076 
0.03982 
0.03889 
0.03796 
0.03704 
0.03611 

0.03519 
0.03427 
0.03335 
0.03243 

56 

11 
g 

61 
62 
63 
64 
65 
66 
67 

0.00612 
0.00696 
0.00780 
0.00863 
0.00946 
0.01029 

O.OIII2 
O.OII95 
0.01278 
0.01360 
0.01443 
0.01525 

*3 

0.03152 

68 

0.01607 

Factor  depending  upon 
the    attached    ther 
mometer. 

14 
15 
16 

\l 

0.03060 
0.02969 

0.02878 
0.02787 

69 
70 

7i 
72 

0.01689 
0.01770 
0.01852 
0.01933 

73 

O.O2OI5 

F. 

LogT. 

19 
20 

0.02606 
0.02516 

74 
75 

0.02096 
O.O2I77 

22 

0.02336 

76 
77 

0.02338 

-30 

+  0.00242 

23 
24 

0.02247 
0.02157 

78 
79 

0.02419 
0.02499 

—  10 

0.00164 

25 
26 

0.02068 
0.01979 

80 
81 

0.02579 
0.02659 

27 

0.01890 

82 

0.02738 

28 

0.01801 

83 

0.02819 

20 
30 
40 

0.00047 
+  0.00008 
—  0.00031 

29 
3» 
31 

0.01713 
0.01624 

0.01536 

84 
85 
S6 

0.02898 
0.02978 
0.03057 

fin 

32 

0.01448 

87 

0.03136 

33 

0.01360 

88 

0.03216 

80 
90 

0.00186 
0.00225 

34 
+  35 

0.01273 
+  0.01185 

89 
+  90 

0.03294 
—  0.03373 

0.00264 

Log  /?  =  log  B  +  log  T. 


LATITUDE. 


237 


LIV.  —  To  determine  the  Latitude  from  the  Meridional  Altitude  of 
an  Object  whose  Decimation  is  known. 

i.  When  the  object  observed  is  south  of  the  zenith  : 
L  =  9o°+D—  A  =  Z+D  =  9o°  +  Z—  A  =  180°  —  (A+  A) 


2.  When  the  star  is  between  the  zenith  and  the  pole  : 

L  =  A  -  A  =  D  -  Z  =  90°  -  (Z  +   A)  =  A  +   D  -  90° 

3.  When  the  star  is  between  the  pole  and  the  horizon  to  the 
north  : 

L  =  A  +  A  =  90°  +  A  —  Z  =  90°  +  A  —  D  =  180°  —  (Z  -f  D) 

where  L  =  the  latitude  sought  ; 

D  =  the  declination  of  the  object,  minus  when  south  ; 

A  =  its  north-polar  distance  ; 
A  =  its  meridional  altitude  ;  and 
Z   =  its  meridional  zenith-distance. 

A  and  Z  must  be  corrected  for  refraction. 
When  the  sun  is  the  object  observed, 
A  =  observed  altitude  —  (refraction  —  parallax)  i  semi-diameter. 


238  ASTRONOMY. 


LV. — Determination  of  the  Latitude  of  a  Place  by  the  Method  of 
Circum- Meridian  A Ititudes. 


Reduction  to  meridian  = 

.  (  .  cos  /  cos  L)  )  ( 

x  =  k  \  i >  —  ;//  tan  a  < 

(         cos  a       J  ( 


.  cos  /  cos  D  )  (  .  cos  /  cos  D 

/ 


cos  a 


2  sin2  \  p 
sin  i" 


2  sin4 
;;/  = 


sin  i 

^o 


a  =  90°  +  D  —  / 
Where— 
A  =  a  -f-  *  =  the  meridional  altitude  of  the  object ; 

a  =  its  observed  altitude  —  (refraction  —  parallax)    i    semi- 
diameter  ; 

p   =  its  correct  hour-angle  ; 

D  =  its  declination ; 

/    =  the  assumed  latitude  of  the  place ;  and 

x   =  the  required  correction  in  seconds. 

When  a  star  is  the  object  observed  and  the  chronometer  marks 

mean  time — 

/  =  1.005473;  log  /  =  0.0023708 

When  the  sun  is  observed  and  the  chronometer  marks  sidereal 

time — 

*  =  0.99455418;  log  /  =  9.9976285 

and,  generally,  when  the  chronometer  has  a  large  losing  rate,  x 
must  be  multiplied  by  i  +  0.00002315  r\  when  it  has  a  gaining 
rate  it  must  be  divided  by  i  -f-  00002315  r\  r  being  the  rate  in 
24  hours,  which  must  be  assumed  minus  when  gaining,  and  plus 
when  losing. 


LATITUDE.  239 


LV. — Determination  of  the  Latitude,  &c.  —  Continued. 

The  values  of  k  and  m  for  each  value  of  p  are  given  in  the 
following  tables. 

The  meridian  altitude, 

A  =  a  4-  x 

for  each  observation ;  for  any  number  of  observations,  n, 


a'  4.  gn  4.    .    .    .    .        x1  +  x"  + 


=  the  mean,  a,  of  all  the  observed  altitudes  -f-  the  mean,  x,  of 
all  the  corrections.     Consequently, 

1.  Measure  several  successive  altitudes   of   the  object   both 
before  and  after  its  meridional  passage. 

2.  Note  the  times  of  each  observation,  and  compute  the  time 
of  the  object's  culmination  ;  the  differences  between  this  and  the 
times  of  each  successive  observation  are  the  values  of  /',  pn  ', 
&c.,  in  time,  for  which  the  corresponding  values  of  k'  ,  k",  &c., 
and  m',  m",  &c.,  must  be  taken  from  the  tables. 

3.  The  means  k  and  m  of  these  results  will  be  introduced  into 
the  equation  for  the  value  of  the  correction,  x,  to  be  applied  to 
a  to  obtain  the  meridional  altitude,  A,  of  the  object. 

4.  If  the  final  latitude  differ  much  from  the  assumed,  the  com 
putation  should  be  repeated  with  the  new  value  for  /. 

5.  It  is  not  necessary  that  the  time  of  the  object's  culmination 
should  be  known  with  great  precision,  provided  an  equal  number 
of  altitudes  be  taken  upon  each  side  of  the  meridian,  and  at 
nearly  equal  distances  from  it. 

6.  The  second  correction,  m,  is  seldom  necessary,  unless  great 
accuracy  is  desired,  and  the  object  is  observed  more  than  ten 
minutes  of  time  from  the  meridian. 


—  -  .  

240                                                       ASTRONOMY. 

t 

Reduction  to  the  Meridian  ;  Values  of  k  —  —  ,  —  f-^_ 

sin  i 

Sec. 

om 

Im 

2m    • 

3m 

4™ 

5m 

6™ 

7m 

u 

// 

// 

u 

u 

// 

u 

II 

0 

0.00 

I.96 

7.8 

17.7 

3i-4 

49.1 

70.7 

96.2 

i 

0.  CO 

2.03 

8.0 

17.9 

31.7 

49-4 

71.1 

96.7 

•  2 

0.00 

2.  10 

8.1 

1  8.  i 

31.9 

49.7 

7L5 

97.1 

3 

0.00 

2.16 

8.2 

18.3 

32.2 

50.1 

7L9 

97.6 

4 

0.  01 

2.23 

8.4 

18.5 

32.5 

5°«4 

72.3 

98.0 

5 

0.  01 

2.3I 

8.5 

18.7 

32.7 

50-7 

72.7 

98.5 

6 

0.02 

2.38 

8.7 

18.9 

33-0 

5i.i 

73-1 

99.0 

7 

0.02 

2-45 

8.8 

19.1 

33-3 

5L4 

73-5 

99-4 

8 

0.03 

2.52 

8.9 

19.3 

33-5 

51-7 

73-9 

99-9 

9 

0.04 

2.60 

9-1 

19.5 

33-8 

52-1 

74-3 

100.4 

10 

o.  05 

2.67 

9.2 

19.7 

34-1 

52-  4     74-  7 

100.8 

ii 

•  o.  06 

2-75 

9-4 

19.9 

34-4 

S2.  7     75-  1 

101.3 

12 

0.08 

2.83 

9-5 

20.  i 

34-6 

53-  1     75-  5 

101.8 

13 

o.  09 

2.91 

9.6 

20.3 

34-9 

53-4     75.9 

102.3 

H 

0.  II 

2.99 

9.8 

20.5 

35-2 

53-8     76.3 

102.  7 

15 

O.  12 

3-07 

9-9 

20.7 

35-5 

54-1 

76.7 

103.2 

16 

o.  14 

•  3.15 

10.  I 

20.  9 

35.7 

54-5 

77.1 

103.7 

17 

o.  16 

3-23 

10.2 

21.2 

36.  o 

54-8 

77-5 

104.2 

18 

o.  18 

3-32 

10.4 

21.4 

36.3 

55-1 

77-9 

104.6 

19 

o.  20 

3-40 

10.5 

21.6 

36.6 

55.5 

78.3 

105.1 

20 

0.22 

3-49 

10.7 

21.8 

36.9 

55.8  1  78.8 

105.6 

21 

o.  24 

3-58 

10.8 

22.0 

37-2 

56.2    79.2 

106.  i 

22 

0.26 

3-67 

II.  0 

22.3 

37-4 

56.5    79.6 

106.6 

23 

0.28 

3.76 

1  1.2 

22.5 

37-7 

56.9   80.0 

107.0 

24 

0.3I 

3-85 

ii.  3 

22.7 

38.0 

57.3    80.4 

107-5 

25 

0-34 

3-94 

ii.  5 

22.9 

38.3 

57.6   80.8 

1  08.  o 

26 

°-37 

4.03 

11.  6 

23.1 

38.6 

58.0   81.3 

108.5 

27 

0.40 

4.12 

ii.  8 

23.4 

38.9 

58.3 

81.7 

109.0 

28 

0-43 

4.22 

11.9 

23.6 

39-2 

58.  7  !  82.  i 

109.5 

29 

0.46 

4.32  ;      12.  i 

23-8 

39.5 

59-  o     82.  5 

1  10.  0 

t 

LATITUDE. 


241 


Reduction  to  the  Meridian;    Values  of  k  = 


2  sm 


sin  i 


Sec. 

Om 

,m 

2m 

3" 

5m 

6m 

r 

" 

n 

n 

II 

n 

// 

n 

n 

3° 

0.49 

4.42 

I2.3 

24.O 

39-8 

59.4 

83.0 

110.4 

3i 

0.52 

4.52 

I2.4 

24.3 

40.  i 

59.8 

83.4 

no.  9 

32 

0.56 

4.62 

12.6 

24-5 

40.3 

60.  i 

83.8 

in.  4 

33 

0-59 

4.72 

12.8 

24.7 

40.6 

60.  5 

84.2 

111.9 

34 

0.63 

4.82 

12.9 

25.0 

40.9 

60.8 

84.7 

112.4 

35 

o.  67 

4.92 

13.1 

25.2 

41.2 

61.2 

85.1 

112.  9 

36 

0.71 

5.03 

13.3 

25.4 

4L5 

61.6 

85.5 

H3-4 

37 

o.75 

5-13 

13.4 

25-7 

41.8 

61.9 

86.0 

H3-9 

38 

0.80 

5-24 

13.6 

25.9 

42.1 

62.3 

86.4 

114.4 

39 

0.83 

5-34 

13.8 

26.2 

42.5 

62.7 

86.8 

114.9 

i 

40 

0.87 

5-45 

14.0 

26.4 

42.8 

63.  o 

87.3 

4i 

0.91 

5.56 

14.  1 

26.6 

43-1 

63.4 

87.7 

115-9 

42 

0.96 

5.67 

14.3 

26.  9 

43-4 

63.8 

88.1 

116.4 

43 

I.  01 

5-78 

14.5 

27.1 

43.7 

64.2 

88.6 

116.9 

44 

i.  06 

5-90 

14.7 

27.4 

44.0 

64-5 

89.0 

117.4 

45 

I.  10 

6.01 

14.8 

27.6 

44-3 

64.9 

89.5 

117.9 

46 

I-  r5 

6.13 

15.0 

27-9 

44.6 

65-3 

89.9 

118.4 

47 

1.20 

6.  24 

15.2 

28.1 

44-9 

65-7 

90.3 

118.9 

48 

1.26 

6.  36 

15.4 

28.3 

45-2 

66.0 

90.8 

H9.5 

49 

'•SI 

6.48 

15.6 

28.6 

45-5 

66.4 

91.2 

120.0 

50 

I.36 

6.60 

15.8 

28.8 

45.9 

66;8 

91.7 

120.5 

5i 

1.42 

6.72 

15.9 

29.1 

46.2 

67.2 

92.1 

121.  O 

S2 

1.48 

6.84 

1  6.  i 

29.4 

46.5 

67.6 

92.6 

I2I.5 

53 

L53 

6.96 

16.3 

29.  6 

46.8 

68.0 

93-0 

122.0 

54 

i-59 

7.09 

16.5 

29.9 

47-1 

68.3 

93-5 

122.5 

55 

1.65 

7.21 

16.7 

30.1 

47.5 

68.7 

93-9 

I23.I 

56 

1.71 

7-34 

16.  9 

3°«  4 

47-8 

69.1 

94-4 

123.6 

57 

1.77 

7.46 

17.1 

30.  6 

48.  i 

69.5 

94-8 

124.  I 

58 

1-83 

7.60 

17.3 

30.9 

48.4 

69.9 

95-3 

124.6 

59 

1.89 

7-72 

17-5 

3i.  i 

48.8 

70.3 

95-7 

I25.I 

16 


242                                                         ASTRONOMY. 

2  sin   ^  i) 
Reduction  to  the  Meridian  ;    Values  of  k  =  —  .  — 
sin  i" 

Sec. 

8"' 

9m 

I0m  ' 

um 

12™ 

i3m 

14'" 

n 

n 

n 

n 

n 

// 

•n 

o 

125.7 

159.0 

196.3 

237.5 

282.  7 

33L8 

384-  7 

i 

126.2 

159.6 

197.0 

238.3 

283.5 

332.6 

385.6 

2 

126.  7 

160.2 

197.6 

239.0 

284.2 

333-4 

386.6 

3 

127.2 

160.8 

198.3 

239.7 

285.0 

334-3 

387.  5 

4 

127.8 

161.4 

198.9 

240.4 

285.8 

335-2 

388.4 

5 

128.3 

162.  o 

199.6 

241.2 

286.6 

336.0 

389-3 

6 

128.8 

162.6 

200.  3 

241.9 

287.4 

336.9 

390.2 

7 

129.3 

163.  2 

200.  9 

242.6 

288.2 

337-7 

391-1 

8 

129.9 

163.8 

201.6 

243-3 

289.0 

338.6 

392.1 

9 

130.4 

164.4 

202.2 

244.1 

289.8 

339-4 

393-0 

10 

131.0 

165.0 

202.9 

244.8 

290.  6 

340.3 

393-9 

ii 

131-5 

165.6 

203.6 

245-5 

291.4 

341-2 

394.8 

12 

132.0 

166.2 

204.2 

246.3 

292.2 

342.0 

395-8 

13 

132.6 

166.8 

204.9 

247.0 

293.0 

342.9 

396.7 

14 

i33.i 

167.4 

205.6 

247.7 

293.8 

343-7 

397-6 

15 

!33.6 

168.0 

206.  3 

248.5 

294.6 

344-6 

398.6 

16 

134.2 

168.6 

206.  9 

249.2 

295-4 

345-5 

399-  5    ' 

17 

134.7 

169.2 

207.  6 

249.9 

296.2 

346.4 

400.5 

18 

135.3 

169.8 

208.3 

250.7 

297.0 

347-2 

401.4 

19 

135.8 

170.4 

208.  9 

251.4 

297.8 

348.1 

402.3 

20 

136.3 

I7I.O 

209.6 

252.2 

298.6 

349-0 

403-3 

21 

136.9 

I7I.6 

210.3 

253.0 

299.4 

349-8 

404.2 

22 

137.4 

172.2 

211.  0 

253.6 

3OO.  2 

350.7 

405.1 

23 

138.0 

172.9 

211.  7 

254-4 

301.  o 

351.6 

406.  o 

24 

138.5 

173.5 

212.3 

255-1 

301.8 

352.5 

407.0 

25 

I39.1 

I74.I 

213.  o 

255.9 

302.  6 

353-3 

408.  o 

26 

139.6 

174-  7 

213.7 

256.6 

303.5 

354-2 

408.9 

27 

140.2 

175-3 

214.4 

257.4 

3°4-3 

355-1 

409.9 

28 

140.7 

175-9 

215.  I 

258.  I 

305-1 

356.o 

410.  8 

29 

141-3 

176.6 

215.8 

258.9 

305.9 

356-9 

411.  7 

1 

LATITUDE. 


243 


Reduction  to  the  Meridian  •    Values  of  k  = 

J 


%  P 


sin  i 


Sec 

gm 

9m 

I0m 

II™ 

I2ln 

I3m 

14™ 

u 

// 

n 

// 

// 

n 

// 

3° 

141.8 

177.2 

216.4 

259.6 

306.7 

357.7 

412.7 

31 

142.4 

177.8 

217.  i 

260.4 

307.5 

358.6 

413.6 

32 

143.0 

178.4 

217.8 

261.  i 

308.4 

359-5 

414.6 

33 

143-5 

179.0 

218.5 

261.  9 

309.2 

360.4 

415.5 

34 

144.1 

179.7 

219.2 

262.6 

310.0 

36L3 

416.5 

35 

144.6 

180.3 

219.9 

263,4 

310.8 

362.2 

417-5 

36 

145.2 

180.9 

220.  6 

264.  i 

3".6 

363.1 

418.4 

37 

145.8 

181.6 

221.3 

264.9 

312.5 

364.0 

419.4 

38 

146.3 

182.2 

222.0 

265.7 

3I3.3 

364.8 

420.3 

39 

146.9 

182.8 

222.  7 

266.4 

3H.I 

36<5.7 

421.3 

* 

- 

40 

147-5 

183.5 

223.4 

267.  2 

3r5-o 

366.6 

422.2 

4i 

148.0 

184.  i 

224.  I 

267.9 

315.8 

367.5 

423.2 

42 

148.6 

184.7 

224.8 

268.7 

316.6 

368.4 

424.2 

43 

149.2 

185.4 

225.5 

269.5 

317.4 

369.3 

425-  i 

44 

149-7 

1  86.  o 

226.  2 

270.3 

318.3 

370.2 

426.  i 

45 

150.3 

186.6 

226.  9 

271.0 

3i9.i 

37LI 

427.0 

46 

IS0^ 

187.3 

227.  6 

271.8 

319.9 

372.0 

428.  o 

47 

I5L5 

187.9 

228.3 

272.6 

320.8 

372.9 

429.0 

48 

152.  o 

188.5 

229.  o 

273-3 

321.6 

373-8 

429.9 

49 

152.  6 

189.2 

229;  7 

274.1 

322.4 

374-7 

430.9 

50 

153-2 

189.8 

230.4 

274.9 

323-3 

375-6 

43L9 

5i 

153.8 

190.5 

231.1 

275.6 

324-1 

376.5 

432.8 

52 

154.4 

191.  i 

231.8 

276.4 

325-0 

377-4 

433-8 

53 

154.9 

191.8 

232.5 

277.2 

325-8 

378.3 

434-8 

54 

155-5 

192.4 

233.2 

278.0 

326.7 

379.3 

435-8 

55 

156.  i 

I93.I 

234.0 

278.8 

327.5 

380.2 

436.7 

56 

i56.7 

193.7 

234.7 

279.5 

328.4 

381.1 

437-7 

57 

157.3 

194.4 

235-4 

280.3 

329-2 

382.0 

438.7 

58 

157.8 

195.0 

236.  i 

28I.I 

33°.o 

382.9 

439-7 

59 

158.4 

195.7 

236.8 

281.9 

330.9 

383-8 

440.6 

244                                                       ASTRONOMY. 

2  sin2  A   -h 

Reduction  to  the  Meridian  ;  Values  of  k 

£      -L 

sin  i" 

Sec. 

i5m 

i6m 

17™ 

i8«" 

19™ 

20m                    2Im 

„ 

n 

n 

n 

n 

// 

a 

0 

441.  6 

502.5 

567-2 

635.9 

708.4 

784.9 

865-3 

i 

442.  6 

5°3-5 

568.3 

637-0 

709.7 

786.  2             866.  6 

2 

443-  6 

504.6 

569-4 

638.2 

710.9 

787.  5            868.  o 

3 

444.6 

505-6 

57o:5 

639-4 

712.1 

788.8 

^69.4 

4 

445-  6 

506.7 

571-6 

640.  6 

7I3-4 

790.1 

870.8 

5 

446.5 

5n7-  7 

572.8 

641.7 

714.6 

791.4 

872.  i 

6 

447-5 

508.8 

573-9 

642.9 

7I5-9 

792.7 

873-5 

7 

448.5 

509.8 

575-0 

644.1 

717.1 

794-  o  , 

874.9 

8 

449-5 

5io-9 

576.1 

645.3 

718.4 

795-4 

876.3 

9 

450-5 

5»-9 

577-2 

646.5 

719.6 

796.7 

877.6 

10 

451-5 

S13>° 

578.4 

647.7 

720.9 

798.  o            879.  o 

ii 

452-5 

514.0 

579-5 

648.9 

722.  i 

799-  3            880.  4 

12 

453-5 

5i5.i 

580.6 

650.  o 

723.4 

800.7 

881.8 

13 

454-5 

516.  i 

581.7 

651.  2 

724.6 

802.0 

883.2 

14 

455-5 

5*7.2 

582.9 

652.4 

725.9 

803-3 

884.6 

'5 

456.5 

518-3 

584.0 

653-6 

727.2 

804.6 

886.0 

16 

457-5 

519-3 

585-1 

654.8 

728.4 

806.0 

887.4 

17 

458.5 

520.4 

586.2 

656.0 

729.7 

807.3 

888'.  8 

18 

459-5 

521.5 

587.4 

657.2 

.  730.  9 

808.6 

890.2 

19 

460.5 

522.5 

588.5 

658.4 

732.2 

809.9 

891.6 

20 

461.5 

523-  6 

589.6 

659.6 

733-5 

811.3 

893.0 

21 

462.5 

524.6 

590.8 

660.8 

734-7 

812.6 

894.4 

22 

463-5 

525.7 

59i-9 

662.0 

736-0 

813.9 

895.8 

23 

464-5 

526.8 

593-0 

663.2 

737-3 

815.2 

897.2 

24 

465-5 

527.9 

594-2 

664.4 

738.5 

816.6 

898.6 

25 

466.5 

528.9 

595-3 

665.6 

739-8 

817.9 

900.  o 

26 

467-5 

53o.o 

596.5 

666.8 

741.  i 

819.2 

901.4 

27 

468.5 

53I-I 

597-6 

668.0 

742.3 

820.5 

902.8 

28 

469-5 

532.2 

598.7 

669.2 

743-6 

821.9            904.2 

29 

470.5 

533-2 

599-9 

670.  4 

744-9 

823.2            905.6 

LATITUDE. 


245 


' 

Reduction  to  the  Meridian  ;    Values  of  k  —  —  g(^~pr~ 

Sec. 

i5m 

i6m 

i7m 

i8m 

I9m 

20m 

2Im 

// 

a 

a 

a 

ii 

u 

u 

30 

471-5 

534-3 

601.  o 

671.6 

746.2 

824.6 

907.0 

31 

472.6 

535-4 

6O2.  2 

672.8 

747-4 

825.9 

908.4 

32 

473-  6 

536.5 

603.3 

674.1 

748.7 

827.3 

909.8 

33 

474.6 

537-6 

604.5 

675.3 

750.0 

828.6 

9II.2 

34 

475-6 

538.7 

605.6 

676.5 

751-3 

829.9 

912.  6 

35 

476.6 

539.7 

606.8 

677.7 

752.6 

831.2 

914.0 

36 

477.6 

540.8 

607.9 

678.9 

753-8 

832.6 

915.5 

37 

478.7 

541-9 

609.  I 

680.1 

755-1 

833.9 

916.9 

38 

479-7 

543-0 

610.  2 

681.3 

756.4 

835.3 

918.3 

39 

480.7 

544-1 

611.4 

682.6 

757.7 

836.6 

919.7 

40 

481.  7 

545-2 

612.5 

683.8 

759.o 

838.0 

921.1 

4i 

482.8 

546.3 

613.7 

685.0 

760.2 

839.3 

922.5 

42 

483.8 

547  4 

614.8 

686.2 

761.5 

840.7 

923-9 

43 

484.8 

548.4 

616.  o 

687.4 

762.8 

842.0 

925-3 

44 

485-8 

549-5 

617.2 

688.7 

764.1 

843.4 

926.8 

45 

486.9 

550.6 

618.3 

689.9 

765-4 

844-7 

928.2 

46 

487.9 

55L7 

619.5 

691.  i 

766.7 

846.  I 

929.6 

47 

488.9 

552.8 

620.6 

692.4 

768.0 

847.5 

931  o 

48 

490.0 

553-9 

621.8 

693.6 

769-3 

848.9 

932-4 

49 

491-0 

555.0 

623.0 

694.8 

770.6 

850.2 

933-8 

50 

492.0 

5'56.  i 

624.  i 

696.0 

771.9 

851.6 

935-2 

5i 

493-1 

557-2 

625.3 

697.3 

773-1 

852.9 

936.6 

52 

494.1 

558.3 

626.5 

698.5 

774-5 

854.3 

938.1 

53 

495-2 

559.4 

627.6 

699.7 

775-8 

855.7 

939-5 

54 

496.2 

560.5 

628.8 

701.  o 

777-1 

857.1 

940.9 

55 

497.2 

561.6 

630.  o 

7O2.  2 

778.4 

858.4 

942.3 

56 

498  3 

562.7 

631.2 

703.5 

779-7 

859.8 

943.8 

57 

499-3 

563-9 

632.3 

704.7 

781.0 

86I.I 

945.2 

58 

500.3 

565-0 

633.5 

705.9 

782.3 

862.5 

946.6 

59 

501.4 

566.1 

634.7 

707.1 

783-6 

863.9 

948.1 

•    ! 

246                                                        ASTRONOMY. 

Reduction  to  the  Meridian;    Values 

,  7        2  sin2  J 

P 

V      /v                        •             ii 

sm  i" 

Seconds. 

22ra 

23m 

24™ 

Seconds. 

22m 

23m 

24™ 

ii 

ii 

n 

n 

// 

//                 » 

o 

949.6 

1037.8 

1129.9 

30 

993-2 

1083.  3 

1177.5 

i 

95I-o 

1039-  3 

1131.4 

31 

994-7 

1084.  8 

1179.1 

2 

952.4 

1040.  8 

1133-0 

32 

996.2 

1086.  4 

1180.7 

3 

953-8 

1042.  3 

1134.6 

33 

997.6 

1087.  9 

1182.3 

4 

955-3 

1043.  8 

1136.2 

34 

999.1 

1089.  5 

1183.9 

5 

956.7 

1045.3 

H37-8 

35 

1000.  6 

1091.  o 

1185.5 

6 

958.2 

1046.  8 

H39.3 

36 

1  002.  i 

1092.  6 

1187.  1 

7 

959-6 

1048.  3 

1140.9 

37 

1003.  5 

1094.  i 

1188.7 

8 

961.  i 

1049.  8 

1142.5 

38 

1005.0 

1095-  7 

1190.3 

9 

962.5 

1051-3 

1144.  o 

39 

1006.  5 

1097.  2 

1191.9 

10 

963-9 

1052.  8 

1145.6 

40 

1008.  o 

1098.  8 

"93-5 

ii 

965.4 

1054.3 

1147.2 

4i 

1009.  4 

1100.3 

1195.1 

12 

966.9 

1055-9 

1148.8 

42 

1010.  9 

noi.  9 

1196.7 

13 

968.3 

1057.4 

1150.4 

43 

1012.4 

1103.4 

1198.3 

14 

969.8 

1058.  9 

1152.0 

44 

1013.9 

1105.  o 

1199.9 

15 

971.2 

1060.  4 

U53-6 

45 

1015.4 

1106.  5 

1201.  5 

16 

972.7 

1062.  o 

II55-2 

46 

1016.  9 

1108.  i 

1203.  I 

17 

974.1 

1063.  5 

1156.8 

47 

1018.  4 

1109.  6 

1204.  7 

18 

975-5 

1065.0 

H58.3 

48 

1019.  9 

IIII.  2 

1206.  4 

19 

977.0 

1066.  5 

H59.9 

49 

1021.4 

1112.  7 

1208.  o 

20 

978.5 

1068.  i 

1161.  5 

50 

1022.  8 

1114.3 

1209.  6 

21 

979-9 

1069.  6 

1163.  i 

5i 

1024.  3 

1115.8 

I2II.  2 

22 

981.4 

1071.  i 

1164.7 

52 

1025.  8 

1117.4 

1212.  9 

23 

982.9 

1072.6 

1166.  3 

53 

1027.3 

1118.9 

I2I4.5 

24 

984.4 

1074.  2 

1167.9 

54 

1028.  8 

1120.5 

1216.  i 

25 

985.8 

1075.  7 

1169.5 

55 

1030.  3 

1  122.  0 

1217.7 

26 

987.3 

1077.2 

1171.  i 

56 

1031.  8 

II23.6 

1219.4 

27 

988.8 

1078.  7 

1172.7 

57 

1033-3 

1125.  I 

1221.  O 

28 

990.3 

1080.  3 

1  1  74-  3 

58" 

1034.  8 

1126.  7 

1222.  6 

29 

991.8 

1081.8 

II75-9 

59 

1036.  3 

1128.3 

1224.  2 

LATITUDE. 


247 


Second  Part  of  the  Reduction  to  the  Meridian. 

r  2  sin4  ^  p 

Values  of  m  =  — . — -^-±- 
sm  i" 


Minutes. 

0s 

10s 

20s 

30s 

40* 

jo- 

5 

O.  01 

0.  01 

ii 
O.  OI 

O.  OI 

0.  OI 

O.  OI 

6 

O.  OI 

0.  01 

O.  OI 

o.  02 

o.  02 

o.  02 

7 

o.  02 

0.02 

o.  03 

0.03 

o.  03 

o.  04 

8 

o.  04 

o.  04 

o.  05 

0.05 

o.  05 

o.  06 

9 

o.  06 

0.07 

0.08 

0.08 

0.08 

o.  09 

10 

0.09 

O.  IO 

0.  II 

O.  II 

0.  12 

0.13 

ii 

0.14 

0.15 

0.15 

o.  16 

0.17 

o.  18 

12 

o.  19 

o.  20 

0.22 

0.23 

0.24 

0.25 

13 

0.27 

0.28 

0.30 

0.31 

0-33 

o.34 

H 

o.  36 

0.38 

o  39 

.0.41 

0.43 

o.45 

15 

0.47 

0.49 

0.52 

0-54 

0.56 

o-59 

16 

o.  61 

o.  64 

o.  67 

0.69 

0.72 

o.75 

17 

0.78 

o.  Si 

0.84 

0.88 

o.  91 

0-95 

18 

0.98 

I.  02 

1.  06 

1.09 

1.13 

1.  18 

19 

1.22 

1.26 

1.30 

1-35 

I.  40 

1-44 

20 

1.49 

1.54 

1.  60 

1.65 

1.70 

1.76 

21 

1.82 

1.87 

1-93 

1.99 

2.06 

2.  12 

22 

2.19 

2.25 

2.32 

2.39 

2.  46 

2-54 

23 

2.61 

2.69 

2-77 

2.85 

2-93 

3.01 

24 

3.10 

3.18 

3.27 

3-36 

3-45 

3-55 

25 

364 

3-74 

3.84 

3-94 

4.05 

4.i5 

26 

4.26 

4-37 

4.48 

4.  60 

4.72 

4.83 

27 

4.96 

5.08 

5.  20 

5-33 

5.46 

5.60 

28 

5-73 

5-87 

6.  01 

6.15 

6.  30 

6.44 

29 

6-59 

6.75 

6.  90 

7.06 

7.22 

7.38 

3° 

7-55 

7.72 

7.89 

8.06 

8.24 

8.42 

31 

8.61 

8-79 

8.98 

9.17 

9-37 

9-57 

32 

9-77 

9-97 

10.  18 

10.39 

10.  61 

10.82 

33 

ii.  04 

11.27 

11.50 

u-73 

11.96 

12.  2O 

34 

12.44 

12.  69 

12.94 

13.19 

13-45 

I3.7I 

35 

13-97 

14.24 

14-51 

14.78 

15.  06 

15.35 

248                                                      ASTRONOMY. 

FORM  FOR 

SURVEY  OF  v  DETERMINATION  OF  THE  LATITUDE, 

and  South  of 

DATE  AND  STATION.  —  1843,  October  13.  —  Month  of  the  Big  Black  River, 

NAME  OF  STAR,  y  Pegasi,  South  of  the  Zenith. 

(  Sextant   No.   2197,  by    Troughton    6°  Simms,    and 
INSTRUMENTS  .  .  .  •? 
(  Mean    Solar    Chronometer    No.    76,    by    Charles 

^o; 

Times  of  ob 

MERIDIAN  DISTANCES, 

Q  ^ 

!~ 

^ 

servation 

=  p. 

o    v 

~  fl  h* 
*   Q  .2    || 

S-i       O 

£    g 

by   chro 

In  mean 

In  sidereal 

sin  i" 

->•  o 

•~     ^     J-s 

6 

nometer. 

solar  time. 

time. 

8° 

1       ^       ^ 

h.  m.     s. 

m.    s. 

m.    s. 

n 

'           » 

I 

10  18  40.4 

9  44-2 

9  45-8 

187.3 

3  49-8 

2 

19  44.4 

8  40.2 

8  41.6 

148.3 

2  5^9 

3 

20  48 

7  36.5 

7  37-7 

114.2 

2    27.3 

4 

21  46.4 

6  38.2 

6  39-3 

86.9 

I    47.6 

5 

22   44.4 

5  40-2 

5  4i.  i 

63-4 

£r 

I    I7.8 

6 

23  54 

4  30-5 

4  3i-2 

40.1 

M 

o  49.2 

7 

25    12 

3  12.6 

3  13-  ! 

20.3 

QJ 

'D-, 

o  24.9 

8 

26  46 

i  38.6 

i  38.8 

5-2 

o  06.3 

9 

28  16.4 

o  08.2 

o  08.2 

o.  o 

a 

0   00.0 

10 

29  42 

i  17-4 

i  17.  6 

3.2 

'c 

o  03.  9 

ii 

31  42 

3  17.4 

3  17.9 

21.4 

1/5 

o  26.2 

12 

32  54.4 

4  29.8 

4  3°-5 

40.0 

6 

o  49 

13 

34  18 

5  53-4 

5  54-3 

68.5 

i  24 

J4 

36  14.2 

7  49-6 

7  50.9 

123.5 

2   31-5 

15 

3832.2 

10  07.6 

10  09.2 

202.3 

4  08.2 

16 

40  06 

ii  41-4 

ii  43-3 

269.9 

.  5  31-1 

Observer,  Major  J.  D.  Graham. 

Computer,      do.                do. 

LATITUDE. 


249 


RECORD  AND  COMPUTATION. 

from  Observed  Double  Circum- Meridian  Altitudes  of  Stars,  North 
the  Zenith. 

a  tributary  to  the  river  Saint  JoJin,  Maine. 


Artificial  Horizon  of  Mercury. 

Young. 


True  circum-meri 

Observed  double 
circum-meridi 
an  altitudes  of 

dian    altitude   of 
star,  as  corrected 
for  refraction  and 

True  meridian  al 
titudes  deduced, 

Latitude    deduced 
from     each     ob 
servation,  =  L  = 

star1. 

errors  of  instru 

=(a  +  *)  =  A. 

(90°  -f-  D—  A). 

ment,  =  a. 

o          /         // 

0          ,          „ 

0              /              // 

o          /          // 

H4  34  15 

57  1838.5 

57  22  '28.  3 

46  56  42.  55 

36  15 

57  19  38.5 

57  22  30.4 

56  40.  45 

37  10 

57  20  06 

57  22  33.3 

56  37-55 

38  10 

57  20  36 

57  22  23.6 

56  47-  25 

39  30 

57  21  16 

57  22  33.8 

56  37.05 

40  30 

57  21  46 

57  22  35.2 

56  35.  65 

41  05 

57  22  03.5 

57  22  28.4 

56  42.  45 

41  5° 

57  22  26 

57  22  32.3 

56  34-  55 

4i  5° 

57  22  26 

57  22  26 

56  40.  85 

41  50 

57  22  26 

57  22  29.9 

56  36.  95 

41  oo 

57  22  oi 

57  22  27.2 

56  39.  65 

39  45 

57  21  23.5 

57  22  12.5 

56  58.35 

38  40 

57  20  51 

57  22  15 

56  55-  85 

36  30 

57  19  46 

57  22  17.5 

56  53-  55 

33  20 

57  18  ii 

57  22  19.2 

56  51.85 

3°  5° 

57  16  56 

57  22  27.  i 

46  56  39-  75 

LATITUDE — Deduced  from  a  mean  of   16  altitudes  of  star  y 

Pegasi 46°  56'  43".4 

Deduced  from  a  mean  of  10  altitudes   of  star  y 

Cephei,  observed  this  night  with  same  sextant  ..  46    57  10  .  7 


Mean,  or  latitude  adopted 46°  56'  5  7" 


2  5°  ASTRONOMY. 


Form  for  Record  and  Computation — Continued. 

D=apparentdeclinat'nofstar==i4°i9/io//.85  N.  logcos     9.  98629 
/=  approximate  latitude  of  place  =  46°57/  .  .logcos     9.  83418 


Sum  ...  19.82048 

a = approximate  merid.  alt.  of  star=57°22/io//  .logcos     9.  73176 

cos  /  cos  D 

cos  a         =  constant  multiple  =  1.227  ...  log  o.  08872 

Refraction  (ther.  28°,  bar.  29.14  in.)  for  mean  obs'd  alts.    —  o'  39" 

Index-error  of  sextant _|_  2  40 

*  Error  of  eccentricity,  &c.,  of  sextant -f  i  40 

Apparent  AR.  of  the  star  f  Pegasi ohc>5mi4s.  09 

Sidereal  time  at  mean  noon  at  this  station 13  26   20.  83 

Sidereal  interval  from  mean  noon  of  star's  culmi 
nation  I0  38   53  .  16 

Retardation  of  mean  on  sidereal  time —     i   44  .  96 

Mean  time  of  culmination  of  star  f  Pegasi 10  37    08  .  2 

Chronometer  (C.  Y.  76)  slow  of  mean  time  at  time 

of  observation 08   43  .  6 


Time  by  chronometer  of  culmination  of  star  ^  Pegasi  ioll28m24s.  6 


On  this  night,  October  13,  1843,  Major  Graham 
obtained  for  the  latitude  of  this  station,  from  75 
observations  on  5  stars  south  of  the  zenith,  com 
bined  with  21  observations  on  f  Cephei  and 
Polaris,  to  the  north 46°  56'  56".3 

On  the  night  of  October  24,  by  43  observations  on 
4  southern  stars,  combined  with  2  observations 
on  Y  Cephei,  the  latitude  deduced  was 46  56  57.  2 

On  September  17,  1844,  66  observations  on  north 

and  south  stars  gave  for  the  latitude  of  this  station  46  56  60.4 

*  The  error  of  eccentricity  is  approximately  ascertained  by  comparing 
latitudes,  well  'determined  by  observations  on  north  and  south  stars,  with 
that  which  will  result  from  north  or  south  stars  individually  of  various  me 
ridional  altitudes.  It  varies  with  the  altitudes  observed;  that  is  to  say,  it 
is  different  for  different  parts  of  the  limb  of  the  instrument. 


LATITUDE.  251 


LVI.  —  To  Determine  the  Latitude  by  an  Altitude  of  a  Star  Near 
the  Pole,  at  any  hour. 

L  =  A  —  (A  cos/)  +  «  (A  sin/)2  tan  A  —  /51  (A  sin/)2  (A  cos/) 
where  — 

A  =  the  observed  altitude,  corrected  for  refraction,  &c.  ; 

A  =  the  polar  distance  of  the  star  in  seconds  of  arc; 
a  =  J  sin  i";   log  a  =  4.3845449; 

/9  =  £  sin2  i";  log  ft  =  8.89403;  and 

/  =  the  hour-angle  of  the  star. 

i  /  =  sidereal  time  —  AR.  *  =  solar  time  -j-  AR.  ©  —  AR.  * 

/  is  plus  when  the  star  is  west,  and  minus  when  it  is  east  of  the 
meridian. 

The  sign  of  cos  /  should  also  be  attended  to,  for  when  /  is 
greater  than  6h,  or  90°,  the  cosine  is  negative,  and  the  second 
and  fourth  terms  change  the  sign  minus  to  plus. 

.  The  fourth  term  may  be  generally  omitted  ;  its  greatest  value 
being  only  o"^. 

This  formula  is  only  applicable  to  stars  within  a  very  few 
degrees  of  the  pole. 

For  other  circumpolar  stars  — 

tan  x  =  tan  A  cos  / 


s 


cos  A 


in  which  the  upper  sign  is  used  when  the  star  is  above  the  pole; 
the  under  when  below  the  pole. 


252 


ASTRONOMY. 


FORM  FOR  RECORD 

SURVEY  OF  DETERMINATION  OF  THE 

DATE  AND   STATION. — 1843,    September  6—  Woodstock, 
NAME  OF  STAR. — Polaris,  (a  Ursa  Minoris,)  observed  on 

(  Sextant  No.  2197,  by  Tmtghton  6° 
INSTRUMENTS  ...\ 

Mean  Solar  Chronometer,  No.  2440, 


Times  of  ob- 

MERIDIAN 

DISTANCES. 

o 

servation 

True    sidereal 

1 

by      mean 
solar  chro- 

times  of  ob 

—  A  cos/ 

JE> 

nome  t  e  r 

servation. 

In  sid'l  time, 

In   arc, 

6 

No.  2440. 

P. 

h.  m.    s. 

h.   m.     s. 

h.  m.     s. 

o        /         // 

/     // 

i 

i  33  02.  5 

20  05  34.  i 

4  58  23-  2 

74  35  48 

—  24  18.  i 

2 

i  34  28 

20  06  59.  8 

4  56  57-5 

74  14  22.  5 

—  24  54-  5 

3 

1  35  42.  7 

20  08  14.  7 

4  55  42.  6 

73  55  39 

—  25  19.8 

4 

i  36  38.  2 

20  09  10.  4 

4  54  46.  9 

73  4i  43-  5 

—  25  41-4 

5 

1  39  07.  5 

20  ii  40.  i 

4  S2  17-2 

73  °4  18 

-26  34.7 

6 

I    41    II.  2 

20    13   44.  I 

4  5°  r3-2 

72  33  22.  5 

—  27  27.1 

7 

i  44  28.  2 

20  17  01.  7 

4  46  55-  6 

7i  43  54 

—  28  40.  8 

Observer,  Major  J.  D.  Graham. 
Computer,  Do. 


LATITUDE.                                                            253 

AND  COMPUTATION. 

t 

LATITUDE,  from   Observed  Double  Altitudes  of  Polaris. 

New  Brunswick,  (Grovels  Inn.) 

between  four  and  five  hours  before  its  upper  meridian  passage. 

Simms,  and  Artificial  Horizon  of  Mercury. 

by  Parkinson  d>°  Frodsham. 

-f-  a(  A  sin/)2  . 
tan  A 

.  (A  cos/) 

Observed 
double  alti 
tudes  of  Po 
laris  out  of 
the  merid 
ian. 

True  altitudes 
of  star,  as  cor 
rected  for  re 
fraction    and 
errors  of  in 
strument, 
=  A. 

Latitude   de 
duced  from 
each  obser 
vation, 
=  L. 

i      n 

, 

°     i     n 

0           /                // 

0         /          // 

+    !    !  !  «  63 
+    I    11.41 

-  o.  32 
—  0.33 

93  01.30 
93  02-45 

46  31  58.6 

• 

46  32  36 

46  08  51.8 

46  08  52.  6 

-f-    I    1  1.  2O 

—  0.33 

93  03.  50 

46  33  08.  6 

46  08  59.  7 

+    I    11.04 

—  0.33 

93  04.40 

46  33  33-  6 

46  09  02.  9 

-{-    I    IO.  63 

-  0.34 

93  06.  15  ;      46  34  21 

46  08  56.  6 

-f    I    10.28 

—  o.35 

93  08.  20        46  35  23.  5 

46  09  06.  3 

-f  i  09.68 

—  0.37 

93  I0-  5° 

46  36  38.  5 

46  09  07 

LATITUDE,  deduced  from  a  mean  of  7  altitudes  of  star  Polaris,  46  D  08'  59//-4* 

254 


ASTRONOMY. 


Form  for  Record  and  Computation  —  Continued. 

Apparent  declination  of  star  =  88°  28'  3o".5. 

Apparent  N.  P.  D.  of  star  =  1°  31'  29".$  =  5489".  5  =  A 

Refraction  (ther.  57°;  bar.  30.013  inches)  .........  . 


*  Index-error  of  sextant 


_       «//  . 
* 


Errors  of  eccentricity,  &c.,  of  sextant _j_  z/  2g// 

Apparent  AR.  of  the  star  Polaris  (a  Ursa;  Minoris) i  h  O3m  5  ys. 

Sidereal  time  at  mean  noon  at  this  station :  j    oo     2  j  \ 

Sidereal  interval  from  mean  noon  of  star's  culmination  ... 
Retardation  of  mean  on  sidereal  time .  . , 


10    03     30 .2 

—       2       18.2 
14     OI       12 


Mean  time  of  culmination  of  star  Polaris  ......... 

Chronometer  No.  2440  fast  of  mean  time  at  time  of  observation  .     4    29     24  .8 

Time  by  chronometer  of  culmination  of  star  Polaris  ..........     6h  30™  36s.  8 

The  reduction  of  the  mean  time  of  observation  to  sidereal  time,  in  the  pre 
ceding  example,  might  have  been  omitted  by  using  table  of  AR.  in  Arc  into 
Mean  Time,  pages  198,  &c.  Thus  : 

Mean  time  of  observation 


Mean  time  of  culmination  of  Polaris  ....................       .   6 


O2s 


-,o 


4)1  57111 


Hour-angle,  /,  in  intervals  of  mean  time 

Sidereal  equivalents,  in  arc  ........................   4h    =  60° 

57m  =  14  i?  20  .45 
34s  =  8  31  .40 
o".3  =  4.51 


Computation,  First 

-  74   35  47-75 
Observation. 

ist  term 
logcos/(+)  = 

log        A        = 

A  cos  p 

9.4242480 
3-7395327 

2d 

sin/ 
A 

A  sin/ 

(A  sin/) 
log  a 
tan  A 

term. 
=  9.984" 

=  3-73953 

3d  term. 

.     .     .      =3-16378 

.     .     .       =  7-44728 
log/9       =8.89403 

3.1637807 

24'  18".! 

31'  58".6 

=  3-72364 

ist  term           =  — 
A                     =  46° 

2  =  7.44728 

=  4-38454 
=  0.02325 

46 
2d  term            =  -f- 

07  40  .5 
i   ii  .63 

2d  term  = 

1-85507 
=    7i".63 
+  i'  11".  63 

3d  term=  —  o/7.32 

46 
3d  term            =  — 

08  52  .13 
o  .32 

Latitude           =  46° 

o8'5i".8i 

LATITUDE.  255 


LVI.  —  Determination  of  the  Latitude  by  Transits  over  the  Prime 

Vertical. 

Suppose  a  transit-instrument  so  placed  that  the  transit-  axis  is 
on  the  meridian,  or  very  nearly  so,  and  that  the  axis  is  horizontal, 
and  the  collimation  nothing  : 

i.  Call  the  time  T  at  which  a  star  whose  declination  is  D 
passes  the  middle  wire  of  the  instrument  on  the  eastern  side 
of  the  meridian,  the  clock-correction  to  reduce  the  observed 
time  to  the  true  E,  and  the  right  ascension  of  the  star  AR.  ;  and 
let  T'  and  E'  denote  the  corresponding  quantities  for  the  western 
transit.  Then  the  two  hour  angles,  in  sidereal  time,  will  be,  the 
eastern  negative, 

/  =  T  +  E  -  AR.;  /'  =  T'  +  E'  -  AR. 

Let  the  unknown  latitude  of  the  place  be  L,  and  the  azimuth  of 
the  line  of  collimation  a.  The  spherical  triangle,  formed  by 
great  circles  connecting  the  zenith,  the  pole,  and  the  place  ot 
the  star,  gives  the  following  relations  : 

cos  t  cos  D  sin  L  —  sin  D  cos  L 
cos  D  sin  / 

cos  t1  cos  D  sin  L  —  sin  D  cos  L 

cos  D  sin  /' 
Whence  — 

cos  4    /'       / 


If  the  instrument  is  very  nearly  on  the  prime  vertical, 

cos  J  (f  +  /)  =  cos  o°  =  i 
and  — 

tan  L  =  tan  D  sec  J  (t1  —  t) 

for  the  passage  over  the  middle  wire  of  the  instrument. 

2.  Call  the  time  of  passage  of  the  star,  from  a  side  wire  to  the 
middle  wire,  r. 

Let  the  distance,  in  arc,  of  one  of  the  lateral  wires  from  the 
middle  wire,  measured  on  a  great  circle,  be  15  //  ./  being  the 
equatorial  interval  of  the  wire,  in  time. 


2  5  6  ASTRONOMY. 


LVI. — Prime  Vertical  Transits — Continued. 
Then,  to  reduce  the  transit  over  a  side  wire  to  the  center  wire, 

=  [sin  (L  +  D) .  sin  (L  -  D)  ±  -1//]1 

The  upper  sign  of  the  term  ±  -1/  /  is  to  be  used  for  wires 
crossed  by  the  star  earlier  than  the  middle  wire  in  the  eastern 
transit,  and  later  in  the  western  transit,  and  the  lower  sign  in  the 
opposite  cases.  An  approximate  latitude  may  be  used  for  L. 

3.  Should  the  optical  axis  not  coincide  with  the  middle  wire, 
substitute  /  i   c  for  /  in  the  above,  according  as  the  error  of 
collimation,  c,  lies  on  the  same  or  opposite  sides  of/. 

4.  The  preceding  formula  gives  the  latitude  on  the  supposition 
that  the  axis  of  the  instrument  is  parallel  to  the  horizon.     If  the 
instrument  is  on  the  prime  vertical,  but  the  north  end  of  the  axis 
is,  for  instance,  n  seconds  too  high,  the  axis  is  parallel  to  the 
horizon  of  a  place  whose  latitude  is  n  seconds  less  than  where 
the  instrument  is  placed,  and  the  true  latitude  is,  therefore, 

L  +  n 

5.  But  should  the  instrument  not  be  on  the  prime  vertical,  the 
true  latitude  becomes — 

L  -[-  n  sin  a 

a  being  the  azimuth  of  the  center  wire  of  the  telescope,  supposed 
in  collimation. 

This  may  be  found  from  the  time  elapsed  between  the  east 
and  west  transits  of  the  same  star :  thus — 

cot  u  =  tan  £  (  t'  —  t)   sin  D 
sin  u 


sin  a  =  cos  D 


cos  L 


a  is  taken  between  o°  and  90°  when  the  north  end  of  the  transit- 
axis  is  between  the  north  and  west,  and  between  90°  and  180° 
when  the  same  end  is  between  the  north  and  east. 

If  n  is  called  plus  when  the  north  end  of  the  axis  is  too  high, 
and  vice  versa,  the  signs  of  the  corrections  are  indicated  by  those 
of  the  quantities  resulting  from  the  formula. 

When  a  is  nearly  90°,  the  correction  is  exceedingly  small ;  so 
that,  when  the  instrument  is  placed  nearly  east  and  west,  we 
may  proceed  in  all  the  computations  as  if  it  were  exactly  so. 


LATITUDE.  2  cj7 


LVI. — Prime   Vertical  Transits — Continued. 

6.  The  instrument  should  be  set  up  in  the  firmest  manner.     A 
change  of  azimuth  between  the  east  and  west  transits  of  a  star 
will  affect  the  result  much  less  than  an  equal  change  of  level. 

It  is  better,  in  order  to  obtain  a  close  result  in  the  shortest 
time,  to  observe  several  stars  on  the  same  evening,  and  between 
the  first  and  last  observations  to  determine  with  the  level  the 
inclination  of  the  axis  several  times,  and  then  to  interpolate  for 
transits  between  the  times  of  observation  of  the  level.  It  is  of 
course  understood  that  the  changes  of  inclination  must  be  small, 
which  will  be  the  case  if  the  instrument  is  properly  placed. 

7.  In  order  to  point  the  telescope  rightly,  the  hour-angles  and 
zenith-distances  of  the  stars  to  be  observed  must  be  computed 
for  the  time  of  transit. 

When  the  telescope  is  on  the  prime  vertical,  calling  p  the  hour- 
angle,  and  z  the  zenith-distance  of  the  star,  then — 

cos/  =  tan  D  cot  L 


cos  z  ^=  -r 


An  allowance  must  be  made  for  the  time  of  crossing  the  first 
wire,  and  for  change  of  zenith-distance  from  the  first  to  the 
middle  wire. 

8.  To  correct,  for  errors  of  collimation,  irregularity  in  the 
pivots,  &c.,  the  instrument  may  be  reversed  between  the  transits 
over  each  vertical;  /.  e.,  the  wires  on  one  side  of  the  center  wire 
are  observed,  the  instrument  reversed  in  its  Y's,  and  the  transit 
over  the  same  wires  continued,  but  in  an  inverse  order;  so  that, 
in  each  vertical  the  same  wire  is  at  one  time  as  far  north  as  it  is 
at  another  south  of  the  optical  axis. 
Then  let— 

L  =  the  latitude  sought ; 
D  =  the  apparent  declination  of  the  star; 
t  =  the  hour-angle,  illuminated  axis  north; 
=  J  diff.  of  sidereal  time  of  transit  over  the  same  wire, 

for  same  position  of  axis ;  and 
t1  —  hour-angle,  illuminated  axis  south. 


tan  D 
tan  L  = 


cos  J  (if  +  /)  .  cos  %  (t1  — 


258  ASTRONOMY 


LVII. —  To  determine  the  Latitude  of  a  Place  by  observing  the 
Difference  of  the  Meridional  Zenith-Distances  of  Two  Stars  on 
Opposite  Sides  of  the  Zenith,  with  the  Zenith  and  Equal-Alti 
tude  Telescope. 

Compute  an  approximate  latitude  by  the  formula — 
L  =  J  [180°  -  (  A  +  A')]  +  i  (*  -  *') 

where  A  and  A7  are  the  polar  distances  of  the  south  and  north 
stars,  respectively,  and  (s  —  z')  the  quantity  measured  by  the 
micrometer.  Then — 

1.  The  correction   for   level  is  applied  by   adding  the  angle 
which  the  vertical  axis  of  the  instrument  makes  with  the  zenith 
when  the  inclination  is  southward,  or  subtracting  it  when  to  the 
northward.     This  correction  is  found  by  multiplying  the  value 
of  one  division  of  the  level-scale,  in  arc,  by  one-half  the  mean 
change,  in   level-divisions,  which  any  one  end  of  the  bubble 
undergoes  by  reversing  the  instrument  on  the  meridian ;  or,  if  o 
and  e,  o'  and  <?',  denote  the  readings  of  the  object  and  eye-ends 
of  the  bubble,  for  south  and  north  stars,  respectively ;  corrections 
for  level  =  ^  (o'  —  e'}  —  J  (o  —  c)  x  the  value  of  one  division 
of  the  level-scale  in  arc. 

2.  The  correction  for  error  of  meridional  position  of  the  central 
vertical  wire  is  found  by  computing  the  usual  "reduction  to  the 
meridian  "  for  each  star ;  then  the  difference  between  the  reduc 
tions  for  the  northern  and  southern  stars  is  taken,  and  one-half 
that  difference  added  or  subtracted,  according  as  the  reduction 
for  the  northern  star  is  greater  or  less  than  that  for  the  southern ; 
or, 


correction  for  position,  = 


;//  being  the  reduction  for  stars  south,  and  m1  for  stars  north  of 
the  zenith. 


* 

LATITUDE. 

259 

L  VI  I.  —  Zenith  Telescope  —  Continued. 

3 

.  When  the  star  is  obser 

ved  off  the  line  of  collimation,  the 

instrument  remaining  in 

the  plane 

of  the  meridian, 

1 

2  sin2  ^  i>        .    . 

f 

—  J. 

X  $  sin  2 

D 

sin 

i" 

The  correction  to  the  latitude  is  one-half  of  this  quantity,  whether 

the 

star  be  north  or  south  ;  and  if 

the  two  stars 

forming  a  pair 

are 

observed  off  the  line  of  collimation,  two  such  corrections, 

separately  computed,  must  be 

added  to 

the  latitude.     D  is  minus 

wJien  south. 

Values  of  m  are 

given  in  the  following  table  : 

D. 

10* 

'5' 

20s 

25s 

30- 

35s 

40' 

45s 

50* 

55'     6 

o"      D. 

Q 

5 

.00 

.01 

.02 

•03 

.04 

.06 

.08 

.  10 

.  12 

.14    .1 

"                O 

7        85 

10 

.01 

.02 

.04 

.06 

.08 

.  ii 

•J5 

.19 

•23 

.28     .; 

!4         80 

15 

.01 

•03 

•05 

.09 

.  12 

•  17 

.22 

.28 

•34 

.41     .4 

^9         75 

20 

.02 

.04 

.07 

.  ii 

.16 

.22 

.28 

•36 

•44 

•53     -< 

>3         7o 

25 

.02 

•05 

.08 

•13 

.19 

.26 

•34 

.42 

•52 

•63     -3 

r5        65 

30 

.02 

•05 

.09 

•15 

.21 

.29 

•38 

.48 

•59 

.71     .* 

5        60 

35 

•  03 

.06 

.  IO 

.16 

•23 

•31 

.41 

•52 

.64 

•77     ><> 

2        55 

40 

•  03 

.06 

.  II 

•  i7 

.24 

•33 

•43 

•54 

.67 

.81     .5 

7        5° 

45 

•  03 

.06 

.  II 

•  17 

•25 

•33 

•44 

•55 

.68 

.82     .5 

8        45 

4.  The   correction  for 

refraction 

is  applied  similarly  to 

reduc- 

tion 

to  meridian  (2)  but 

with 

a  contrary 

sign 

;  or, 

Prm 

f*C*\"\  f\ 

r 

—  r1 

„ 

v-xUl  J 

CL'llU 

2 

—  - 

- 

r 

—  r1 

being  small,  no  note  need  be  taken 

of  the  state 

of  the 

barometer  and  thermometer  at  the 

time 

of  observation. 

260                                                         ASTRONOMY. 

L  VI  I.  —  Zenith   Telescope  —  Continued. 

The  following  table  gives  the  correction  to  the  latitude  for 

differential  refraction;  arguments,  one-half  difference  of  zenith- 

distances  on  one  side  and  zenith-distance  on  the  top  : 

Zenith-distance. 

S—S? 

2 

o° 

10° 

2O° 

25° 

30° 

35° 

/       // 

,, 

ii 

„ 

ii 

u 

tl 

0         0 

.  OO 

.  OO 

.  OO 

.  OO 

.  OO 

.00 

o    30 

.  01 

.  OI 

.  OI 

.  OI 

.01 

.  OI 

I 

.  02 

.  02 

.02             .02 

.02 

.  02 

I  30 

.  02 

•  ^3 

.03         .03 

•03 

•03 

2 

.03 

•  ®3 

.04         .04 

.04 

•05 

2      30 

.04 

.04 

.05         .05 

•05 

.06 

3 

•  05 

•05 

.06 

.06 

.07 

.08 

3    30 

.06 

.06 

.07 

.07 

.08 

.09 

4 

.07 

.07 

.08 

.08 

.09 

.  10 

4    3° 

.08 

.08 

.09 

.09 

.  IO 

.  II 

5 

.08 

.09 

.  10 

.  10 

.  II 

•'3 

5    30 

.09 

.  10 

.  10 

.  II 

.  12 

.14 

6 

.  10 

.  10 

.  II 

.  12 

•13 

.15 

6    30 

.  II 

.  II 

.  12 

•  *3 

.14 

.16 

7 

.  12 

.  12 

•13 

.14 

•15 

.18 

7    30 

•!3 

•13 

•H 

.15 

.16 

.19 

8 

•'3 

.14 

•!5 

.16 

.18 

.21 

8    30 

.14 

•15 

.16 

.17 

.19 

.22 

9 

.15 

.16 

.17 

.18 

.20 

•23 

9    30 

.16 

•17 

.18 

.  20 

.21 

.24 

10 

.17 

.18 

.19 

.  21 

•23 

.26 

10    30 

.18 

•19 

.  20 

.  22 

.24 

.27 

II 

.18 

.19 

.  21 

•23 

•25 

.28 

II    30 

.19 

.20 

.22 

.24 

.26 

•30 

12 

.20 

.21 

•23 

•25 

.27 

•31 

The  sign  of  the  correction  is  the  same  as  that  of  the  micro 

meter-difference. 

LATITUDE.  261 


L VI I. — Zenith   Telescope — Continued. 

5.  To  find  the  value  a  of  one  division  of  the  micrometer, 
note  the  time  by  chronometer  of  the  transit  of  Polaris  or  other 
close  circumpolar  star  over  the  movable  wire  placed  vertically 
and  set  successively  before  the  star  for  each  turn  or  half-turn  of 
the  screw.  Then  let  x  be  the  arigular  distance  from  the  meridian 
at  any  reading  of  the  screw;  /,  the  hour-angle  of  the  star  at  the 
same  instant ;  and  D,  its  declination  : 

sin  x  =  cos  D  sin  / 

The  value  of  x  is  computed  for  each  reading,  and  the  differences 
of  these  values  divided  by  the  differences  of  the  corresponding 
micrometer-readings  give  values  for  the  screw. 

A  better  method,  as  it  avoids  displacing  the  micrometer,  is  to 
observe  a  close  circumpolar  star  near  its  elongation,  when  rapidly 
rising  or  falling,  with  but  slight  motion  in  azimuth.  The  level 
should  be  carefully  noted  in  order  to  allow  for  possible  changes, 
and  a  correction  applied  for  differential  refraction. 

The  sidereal  time  of  elongation  and  the  azimuth  of  the  star 
can  be  determined  from  LXI. 

About  40  or  more  minutes  before  elongation,  transits  are  noted, 
the  micrometer  being  set  in  advance  consecutively  by  whole  or 
half  turns  of  the  screw  throughout  its  length.  A  correction  for 
rate  of  chronometer  should  be  applied  if  sensible. 

Let— 

/    =  the  difference  between  the  time  of  observation  and  the 
time  of  elongation  of  the  star;  and 

s"  =  the  number  of  seconds  of  arc  from  elongation  in  thte 
direction  of  the  vertical. 

s"  =  15  cos  D  [t  —  1  (15  sin  i")2  t?>] 
where  /  is  expressed  in  seconds  of  time. 


262 


ASTRONOMY. 


LVII. — Zenith  Telescope — Continued. 

Values  of  i  (15  sin  i")2  /3  for  minutes  of  time  from  elongation 
are  given  in  the  following  table  : 


/. 

Term. 

t. 

Term.  ||       /. 

Term. 

/. 

Term. 

m. 

s. 

VI.      \           S. 

in. 

s. 

in. 

s. 

5 

o.  o 

15 

0.6 

25 

3-o 

35 

8.2 

6 

o.  o 

16 

0.8 

26 

3-3 

36 

8.9 

7 

o.  o 

17 

0.9 

27 

3-7 

37 

9.6 

8 

O.  I 

is  ;   i.i 

28 

4.2 

38 

10.4 

9 

0.  I 

19 

i-3 

29 

4-6 

39 

ii.  3 

10 

0.2 

20 

i-5 

30 

5-i 

40 

12.2 

ii 

0.2 

21 

T       8 

I.  o 

31 

5-7 

4i 

I3.I 

12 

0-3 

22 

2.  O 

32 

6.2 

42 

14,1 

13 

0.4 

23 

2-3             33 

6.8 

43 

I5.I 

14 

o-5 

24 

2.6 

34 

.  7-5 

44 

16.2 

It  is  convenient  to  apply  these  values  to  the  observed  time  of 
noting,  additive  to  the  observed  time  before,  and  subtractive 
after,  either  elongation. 

The  correction  to  be  applied  to  the  observed  times  of  noting 
for  change  of  level  is  given  by  the  formula  — 


where  N0,  S0,  the  north  and  south  readings  for  a  selected  state 
of  level,  N,  S,  the  readings  for  any  other  state,  and  b  the  value  of 
one  division  of  level-scale  in  seconds  of  arc  ;  the  upper  sign  to 
be  used  for  western,  the  lower  sign  for  eastern  elongation. 

After  these  two  corrections  have  been  applied  we  have  in  one 
column  the  readings  of  the  micrometer,  and  in  another  the  cor 
responding  times,  such  as  would  have  been  obtained  if  the  star 
had  moved  uniformly  in  a  vertical  line,  leaving  out  of  considera 
tion  for  the  present  the  change  in  refraction  and  the  rate  of  the 
chronometer. 

Various  methods  of  combination  might  be  adopted  for  the 
determination  of  the  turn  of  the  screw.  That  of  subtracting  the 
values  resulting  from  the  first  operation  from  those  of  the  middle 
one;  next,  those  of  the  second  from  those  of  the  middle  one, 
plus  one,  and  so  on,  is  recommended  for  its  simplicity. 


LATITUDE.  263 


L VI I . — Zenith   Telescope—  Continued. 

A  number  of  values  are  thus  obtained  for  the  time  of  a  given 
number  of  turns  or  half-turns,  from  which  is  deduced  the  value 
of  one  turn :  thus — 

Mean  time  for  one  turn,  ii6s-774    l°g  —  2-°6735 

cos  D         log  =  8.40750 
15          log  =1.1 7609 

One  turn 44"- 765  —  1-65094 

Correction  for  refraction —  .025 

Correction  for  rate —  .003 


Resulting  value  .......  44  .737 

The  correction  for  refraction,  in  seconds  of  arc,  is  negative  for 
either  eastern  or  western  elongation,  and  equals  the  change  ot 
refraction  for  the  space  equal  to  one  turn  ;  equal  to  the  value  ot 
one  turn  times  the  difference  of  refraction  for  i'  at  star's  altitude 
divided  by  60. 

6.  The  value  b  of  i  division  of  the  level-scale  will  be  best 
found  by  using,  in  conjunction  with  the  micrometer,  a  distant 
point  as  a  mark,  or  the  central  wire  of  another  instrument  used 
as  a  collimator  ;  for  the  space  above  or  below  the  mark,  passed 
over  by  the  horizontal  wire  of  the  micrometer,  during  the  bubble's 
run  over  the  scale,  as  the  telescope's  elevation  is  gradually 
altered,  may  afterward  be  measured  by  the  micrometer-screw. 
The  temperature  should  be  noted,  since  the  result  may  change 
with  a  change  of  temperature. 

Including  all  corrections,  the  general  expression  for  latitude  is  — 


__  .  .,  ,  .  , 

2     ~  2  4  2  "  2 

a  and  b  being  the  arc  values  of  one  division  of  the  micrometer 
and  level-scale  respectively. 

To  correct  as  much  as  possible  an  erroneous  determination  of 
the  value  of  the  micrometer-screw,  select  stars  for  observation, 
such,  if  practicable,  that  the  greatest  zenith-distance  of  a  pair 
will  belong  as  often  to  the  north  star  as  to  the  south  star  ;  be 
cause  if  the  zenith-distance  of  the  north  star  is  the  greatest,  the 
observed  quantity  is  subtractive;  if  least,  additive.  For,  as  a 
general  rule,  the  error  of  latitude  arising  from  an  erroneous  value 
to  the  micrometer-screw  will  be  the  least  when  in  a  set  of  stars  — 


264 


ASTRONOMY. 


b   -a 


cfl 

oi 

1 
1 

1 

S"                  "*•                  ^                  o         oo 
^                                         °°.                      oo                      oo            oo 

o 

v          q             q             S             S 

0066 

cr! 

G 

.2 

2 

000 

o'                   o'                   o'                  6 

III? 

1 

0 

0 

"3 

ro                         r^                        \o                          r^ 

i-1                           o'                          0*                          M 
+                          +                           1                           + 

1 

JJ                         0;                        co                          M               G 
O 

^ 

5 

T~i~s      ?   s      3"   £     s   a? 

i 
i 
i 

i 
i 

3      S 
3      3 

i 

5 

o>      •<*• 

c 
o 

,  i  |    !  ?   1  ?    s  ? 

c 
"o 

Q 

^       m      oo 

0          00             O> 

10       n 

^         f?           H"                                   2                                     ON                 00             ON 

*l 

0                     vo                      M                      m 

+                     +                      1                      + 

00 

ro      q           M       M           O_N      *•         vo       ON 

w         00                VO         vo'                 ON       oo'                 ro         4- 

* 

t--vo             woo            ooro            t^ro 

ONm                t^c^                t^O                  ONCTN 
HW               MM               Mro              COM 

ometer. 

Q 

N 

5 

^                 ON                       ro                       M 
^                     oo                      oo                       ON 
f^'             o 

1 

o 

i 

be 

q 

1 

j-^NM                  fOM                 -^-00                 rOM 
^ONvo^           oo^rl             •*       '0             roin 
Si      2         JJ              2S              M"              M" 

fc     c* 

S5     oi        ^     c/i        ^     c/5        <H     i/J 

5 

O 
o        • 
5?     ^ 
S3 

°3           ON 
VO         vo 

AZIMUTHS.  265 


LVIII. — Knowing  the   Time  and  the  Latitude  of  the  Place,  to 
find  the  Azimuth  of  the  Sun  or  a  Star. 


A  -  J  (A  +  S)  T  J  (A  -  S) 
the  upper  or  negative  sign  is  used  when  A  is  greater  than  A ; 

where 

A  =  the  azimuth  counted  from  the  north,  which  must  be 
subtracted  from  180°  if  counted  from  the  south; 

S   =  the  angle  at  the  star,  called  the  angle  of  variation ; 
X   =  the  co-latitude  of  the  place ; 
A  =  the  north-polar  distance  of  the  sun  or  star ;  and 
p  =  the  hour-angle  at  the  pole. 

Without  the  Use  of  a  Chronometer,  by  observing  the  Altitude  of  the 
Sun  or  Star  at  the  Same  Instant  with  the  Observation  of  the 
Azimuth. 

Let  Z  =  the  zenith-distance,  corrected  for  refraction,  parallax, 
and  semi-diameter:  then — 


sin  k  .  sin  (k  —  A) 

cos2  A  A  =  —    — : — ^-A — '. 

sin  Z  sin  X 


2  k  =  Z  +  A  +  A 


266  ASTRONOMY. 


LIX. — To  find  the  AMPLITUDE  of  a  Celestial  Object  at  its  Rising 
or  Setting;  by  "  Amplitude"  is  meant  the  complement  of  the 
azimuth,  or  distance  from  the  east  or  west  points  of  the  horizon. 

This  is  a  particular  case  of  the  preceding  problem.  When 
the  object  appears  to  be  in  the  horizon,  its  zenith-d.istance,  instead 
of  being  90°,  is,  on  account  of  refraction  and  parallax,  90°  -f  /', 
where — 

k  —  horizontal  refraction  —  horizontal  parallax 
=  36'  29"  —  horizontal  parallax. 

For  stars,  the*  horizontal  parallax  =  o  and  k  =  90°36/  29"  ; 
for  the  sun,  k  =  36/29//  —  8".8  =  90°  36'  2o".2.  The  mean 
refraction  and  mean  horizontal  parallax  are  here  used,  as  these 
observations  are  not  susceptible  of  much  accuracy. 


LX. — To  find  the  True  Meridian  by  the  Method  of  Equal  Alti 
tudes  of  the  Sun. 

The  instrument  remaining  stationary,  observe  the  readings  of 
the  horizontal  limb  when  the  altitude  of  the  sun's  center  is  the" 
same  in  the  forenoon  and  afternoon. 

Then,  the  correction  to  the  mean  of  these  two  readings  for  the 
change  in  the  sun's  declination  in  the  interval  is — 

4  (D  -  D') 


cos  L  .  sin  J  (/  —  f) 

where — 

D  —  D'  =  the  change  in  the  sun's  declination  in  the  interval 
of  the  observations ; 

(t  —  /')  =  this  interval  of  time  expressed  in  arc ;  and 
L  =  the  latitude  of  the  place. 


AZIMUTHS.  267 


LXI. — To  find  the  Azimuth  of  POLARIS,  or  other  Close  Circumpolar 
Star,  at  its  Greatest  Eastern  or  Western  Elongation. 

cos/  =  tan  A  cot  /  =  cot  D  tan  L  =  tan  L  tan  A 

cos  L  sin  A  =  sin  A  =  cos  D 
where — 

p    =  the  hour-angle  of  the  star ; 

D  =  its  declination ; 

A  =  its  polar  distance ; 

A  =  the  required  azimuth ; 

L  =  the  latitude  of  the  place;  and 

I    =  the  co-latitude. 

The   first   equations   give   the  hour-angle  of  the  star  at  its 
greatest  elongation;  hence  the  sidereal  time  of  elongation; 
The  second,  the  azimuth  of  the  star  at  its  greatest  elongation. 
The  azimuth  at  any  hour-angle  is  found  by  the  methods  in 
LVIII,  or  by  the  formula — 

A  (in  seconds)  = ^~  <  A  -f-  A2  sin  i"  cos/  tan  L 

+  J  A3  sin2  i"  [(i  +  4  tan2  L)  cos2/  —  tan2  L]  \ 

If  the  hour-angle  is  counted  from  the  lower  culmination,  change 
the  sign  of  the  second  term. 

The  most  approved  method  is  to  observe  a  series  of  azimuths 
of  the  star  about  the  elongation,  say  for  not  more  than  30  minutes 
before  and  after,  and  to  reduce  them  to  the  elongation.  To  do 
this,  compute,  from  the  known  latitude,  the  azimuth  of  the  star  at 
its  greatest  elongation  =  A,  and  call  the  sidereal  time  from 
elongation  /;  the  correction  to  the  azimuth  will  be — 

c  =  (112.5)  f1  sm  I"  *an  A 
log  (112.5)  sin  *"  =  6-7367274 

The  quantities  found  in  the  table  for  "Reduction  to  the 
meridian" — 

/smH/\ 
V      sin  L"  ) 
correspond  very  nearly  to — 

(112.5)  ^  s^n  **-" 
so,  by  entering  the  table  with  the  time  from  elongation,  and 


268 


ASTRONOMY. 


LXI. — To  find  the  Azimuth  of  Polaris,  &c. — Continued. 

multiplying  the  tabular  quantities  by  tan  A,  we  obtain  the  re 
quired  correction  in  seconds  of  arc.  This  will  be  found  a  con 
venient  substitute  for  the  more  rigorous  method. 

The  formula  may  be  separately  applied  to  each  observation,  or 
the  work  may  be  shortened  by  computing  only  the  azimuth 
corresponding  to  the  mean  hour  angle  and  applying  to  it  a 
correction  to  mean  azimuth. 

Let  11  be  the  number  of  observations  on  the  star ;  Ab  the  azi 
muth  corresponding  to  the  mean  hour-angle;  and  let,  also,  r  = 
the  difference  between  the  time  of  any  observation  and  the  mean 
of  the  times;  then,  for  a  circumpolar  star, 

Correction  to  mean  azimuth  =  A:  —  tan  Ar--  zL!^lil 

n         sin  i" 

Example. 

Means  of  the  times  of  observations  by  chronometer  =  3h  48™  i2s  .3. 


Chronometer  time 
of  observation. 

T 

Tab.  quantities, 
(page  240.) 

log  27".  4  =  1.4380 
log  tan  A  —  8.  5  144 

//.    m.       s. 

3    42    3°-  5 
44    08.  o 
45     52.o 

47     15-  ° 

48    59.  o 

5o    34-  o 
52    28.  o 

3    53    5i-5 

1)1.       S. 

4  04.3 

2   30-3 

57.3 
46.7 

2   21.  7 

4  15.7 
5  39-2 

63.7 
32-5 
10.7 
1.8 

I.  2 
II.  0 

35-7 
62.8 

9-  9524 
Correction.  =  —  o".  90 

IS  =  27.4 

Azimuth  of  star  at  elongation..!0  52'  42".  8 
Chronometer  time  of  elongation.  4h  o6m  2OS.  5 
Mean  time  of  obs'n  by  chron.-3h  48™  12s.  3 

log 
log 
log 

t^                      .  6.0774174 

(112.5)  sin  i".  -6.  7367274 
tan  A    8.5162425 

t  — 

i  088".  2  =      i8m   8".  2 

Cor.  =  2i".2  =..   1.3263873 

Azimuth  corresponding  to  mean  hour-angle  =  i°   52'  2i".6 
Correction  to  mean  azimuth =  0.9 


Mean  azimuth 


=  1"    52'    20".7 


AZIMUTHS.  269 


LXI. — To  find  the  Azimuth  of  Polaris,  &c. — Continued. 

Azimuths  are  usually  reckoned  from  the  south  and  in  the 
direction  of  south  to  west.  When  circumpolar  stars  are  observed, 
it  is  more  convenient  to  reckon  from  the  north  meridian.  The 
determination  of  primary  azimuths  supposes  the  local  time  to  be 
known;  for  secondary  azimuths  observations  for  time  and  azimuth 
may  be  made  together.  The  sun  is  only 'employed  in  connection 
with  the  inferior  class  of  azimuths. 

For  the  purpose  of  referring  azimuths  observed  at  night  to  the 
direction  of  any  geodetic  signal,  a  mark  is  set  up,  consisting  of  a 
perforated  box,  (about  3^  of  a  foot  cube,)  through  the  front  face 
of  which  the  light  of  a  bull's-eye  lantern  is  shown,  appearing  of 
about  the  size  and  brilliancy  of  the  star  observed  upon.  The 
distance  of  this  mark  from  the  place  of  observation  is  generally 
determined  by  local  circumstances,  but  should  not,  if  possible, 
be  nearer  than  about  a  mile,  in  order  that  the  sidereal  focus  of 
the  telescope  may  not  require  changing.  For  day  observations 
a  vertical  black  stripe,  of  the  same  width  as  the  aperture,  is 
painted  upon  a  white  wand  placed  vertically  above  it.  If  the 
diameter  of  the  aperture  is  a  quarter  of  an  inch,  it  will  subtend 
at  the  distance  of  a  mile  an  angle  of  a  little  more  than  o/7.8. 
The  horizontal  angle  between  the  mark  and  any  trigonometrical 
station  is  measured  in  connection  with  the  triangulation  by  com 
bining  it  with  all  other  directions  radiating  from  the  station 
observed  from. 

Observations  for  azimuth  are  usually  made  in  sets,  commencing 
with  a  number  on  the  mark,  followed  by  about  an  equal  number 
of  readings  on  the  star  preceded  and  followed  by  level-readings. 
The  instrument  is  then  reversed,  and  the  preceding  operations  are 
repeated  in  the  reverse  order,  the  number  of  readings  on  the 
star  and  mark  being  as  before. 

In  these  observations  the  optical  axis  of  the  telescope  of  the 
theodolite  must  be  made  to  describe  a  truly  vertical  plane. 

If  the  axis  of  the  telescope  is  not  horizontal,  the  correction  to 
the  azimuth  will  be — 

i  -  |  (W  +  w>)  _  (e  _  ef)  \  tan  L 
where  d  =  the  value  of  one  division  of  the  level-scale  in  seconds 


270  ASTRONOMY. 


LXI. — To  find  the  Azimuth  of  Polaris,  &c. — Continued. 

of  arc,  w,  e,  and  «/',  e',  the  west  and  east  readings  of  the  level 
before  and  after  reversal. 

The  circumpolar  stars  a,  3,  A  Ursae  Minoris  and  51  Cephei  are 
those  almost  exclusively  used.  When  d  Ursae  Minoris  and  51 
Cephei  culminate  on  either  side  of  the  pole,  Polaris  is  not  far 
from  its  elongation;  and,  on  the  contrary,  when  the  pole-star 
culminates,  the  other  two  are  not  far  from  their  elongations  on 
either  side  of  the 'meridian.  A  Ursae  Minoris,  from  its  greater 
proximity  to  the  pole,  and  its  small  size,  presents  to  the  larger 
instruments  a  finer  and  steadier  object  than  Polaris. 


LXII. — Correction  for  RUN  in  reading  Microscopes. 

As  it  is  difficult  to  adjust  the  microscopes  so  that  five  revolu 
tions  of  the  micrometer-screw  shall  carry  the  wire  exactly  over 
one  of  the  five-minute  spaces  on  the  limb  of  the  instrument,  (if 
it  be  so  graduated,)  it  is  preferred  to  observe  the  number  of 
revolutions  and  the  part  of  a  revolution  made  by  the  screw  while 
the  wire  passes  over  the  space ;  then,  let — 

;;/  =  the  mean  of 'first  readings ;  that  is,  the  readings  obtained 
by  turning  the  screw  in  the  direction  of  increasing 
numbers  from  the  zero  of  the  comb  ;  and 

m1  =  the  mean  of  second,  or  reverse,  readings ; 

then — 

(mean)  run  =  r  =  m  —  m'  -f-  300 
and — 

300  .  /;/        700  (r  4-  m'  —  300) 

true  (mean)  readme  =  ^—       -  =  ^ ^— 

r  r 

=  the  number  of  minutes  and  seconds 
to  be  added  to  the  degrees  and 
minutes  of  the  limb. 


LONGITUDE. 


271 


LXIII. — Longitude  by  Lunar  Culminations. 

i.  Make— 

/  =  true  longitude  sought ; 

/'  =  approximate  longitude ; 

m  =  observed  change,  in  right  ascension,  of  the  moon's 
bright  limb  between  the  first  meridian  and  that  sought; 
;;/'  =  computed  change  in  same,  by  interpolation ; 

V  =  rate  of  motion,  in  right  ascension,  of  the  moon's 
bright  limb,  when  on  the  meridian  /' ;  and 

I  =  the  constant  difference  between  the  values  of  the  inde 
pendent  variable,  or  arguments,  corresponding  to  the 
consecutive  tabulated  values  of  the  right  ascension 
of  the  moon ; 


then— 


2.  Interpolation. — Take  the  following  scheme,  viz 


I 

F 

A, 

A2 

A3 

A4 

^ 

/'" 

a"1 

b" 

/" 

a" 

c" 

b' 

d' 

/' 

a' 

c1 

e' 

b 

d 

f 

ft 

<*, 

ci 

e, 

tu 

^ 

cn 

d, 

*,„ 

ain 

b" 

In  which  the  column  I  contains  the  independent  variable,  or 
argument,  as  time,  terrestial  longitude,  degrees,  and  the  like ;  F, 
the  value  of  the  function  of  this  variable,  as  found  in  any  set  of 
tables;  Ab  A2,  &c.,  the  first,  second,  &c.,  order  of  differences  of 
these  functions. 


272 


ASTRONOMY. 


LXIII. — Longitude  by  Lunar  Culminations — Continued. 
Make — 

s  =  the  value  of  the  function  corresponding  to  any  value 
ts  between  t1  and  //; 


t  = 


t  // 

1 1       • 


a  — 


e  = 


a'  +  a, 


Then,  according  to  Bessel,  Ast.  Nach.  No.  30 — 


i)  ./.(/-  i)  (/—  2] 
1.2.3.4" 


.  ^  _i_  £c. 


Or,  making — 

A!  =  b 


A,  =  d 


A,  = 


s  =  a'  +  / 


(3) 


A 

.2  1.2.3 


or,  by  the  ascending  powers  of  /, 

s  =  a'  +  A/  +  B/2  +  C/3  +  D/4  +  &c.    .     .     (4) 


LONGITUDE.  273 


LXIII. — Longitude  by  Lunar  Culminations — Continued, 
in  which,  stopping  at  the  fourth  differences, 
A  =      A,   —  AA2  4-  TLAo  - 


.    .    .    .    (5) 


B  =  JA2  -  JA3  -  -JTA4 

C  =  1A3  -  ,VA4 

D  =  JTA4  3 


Also, 

V  =  ~  =  A  +  2  B  /  +  3  C  /2  +  4  D  /3  +  .     .     (6) 

;«'  =  s  -  a1  =  A  t  +  B  /2  +  C  /3  +  D  /4  -f   .     .  (7) 

The  value  of  m  may  be  obtained  either  from  observations  on 
the  two  meridians,  or  by  observations  on  one  and  the  tabulated 
results  under  the  head  of  Moon  Culminations  in  the  Nautical 
Almanac,  which  may  be  used  as  actual  observations  on  the 
meridian  of  the  ephemeris. 

NOTE. — If  the  lunar  tables  were  perfectly  accurate,  the  true  lon 
gitude  given  by  the  observation  would  be  found  at  once  by  com 
paring  the  observed  right  ascension  with  that  of  the  ephemeris. 
There  are  two  methods  of  avoiding  or  eliminating  the  errors  of 
the  ephemeris.  In  the  first,  the  observation  is  compared  with  a 
corresponding  one  on  the  same  day  at  the  first  meridian,  or  at 
some  meridian  the  longitude  of  which  is  well  established.  In  this 
method  the  increase  of  the  right  ascension  in  passing  from  one 
meridian  to  the  other  is  directly  observed,  and  the  error  of  the 
ephemeris  on  the  day  of  observation  is  consequently  avoided; 
but  observations  at  the  unknown  meridian  are  frequently  rendered 
useless  by  a  failure  to  obtain  the  corresponding  observations  at 
the  first  meridian. 

In  the  second  method,  the  ephemeris  is  first  corrected  by  means 
of  all  the  observations  taken  at  the  fixed  observatories  during  the 
semi-lunation  within  which  the  observation  for  longitude  falls. 
The  corrected  ephemeris  then  takes  the  place  of  the  correspond 
ing  observation,  and  is  even  better  than  the  single  corresponding 
observation,  since  it  has  been  corrected  by  means  of  all  the  ob 
servations  at  the  fixed  observatories  during  the  semi-lunation. — 
Chauvenefs  Practical  Astronomy. 


18 


274  ASTRONOMY. 


LXIII. — Longitude  by  Lunar  Culminations — Continued. 
3.  Example. — Let — 

/'  =  4h  55m  5os  west  from  Greenwich, 

and  suppose  the  following  transits  with  a  chronometer  marking 
sidereal  time.  The  error  of  the  time-keeper  is  not  material,  and 
the  transit  is  very  nearly  in  the  meridian,  viz : 

Feb.  18.  £      Geminorum  6h  54™  4is-75 
d      Geminorum  7     10    38 .97 

])'s  first  limb  .. 7''  38'"  o6s.76 

C     Cancri 8    03     06. u 

3)22    08    26.83     7    22    48-943 

o     15     17.817 
Chronometer  rate  +  3s  daily — o  .0318    oh   15™  I7S.7S5 


The  corresponding  observations  at  Greenwich,  as  given  by  the 
Nautical  Almanac,  are : 

Feb.  18.  £     Geminorum  6h  54™  578.4i 
6      Geminorum  7     10    54 .36 

D  's  first  limb  .. ;b  27™  473.66 

£     Cancri 8    03     21.44 


3)22    09     13.21     7    23    04.403    o1'  04™  433.257 


10    34.528 


Next  compute  this  increase  from  Nautical  Almanac.  The 
right  ascensions  of  the  moon  are  given  in  that  work  for  the  upper 
and  lower  passages  over  the  meridian  of  Greenwich.  The 
independent  variable  is,  therefore,  terrestrial  longitude,  of  which 
the  unit  is  one  hour,  and  the  intervals  between  the  consecutive 
tabulated  values  of  its  function,  12.  The  increase  to  be  com 
puted  is  for  the  interval  of  passage  from  the  upper  meridian  of 
Greenwich  to  that  4h  55™  5oa  west.  Now,  according  to  the 
scheme  and  equations  (2),  (3),  and  (5), 


LONGITUDE. 


275 


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276  ASTRONOMY. 


LXIII. — Longitude  by  Lunar  Culminations — Continued. 

Then  equation  7 — 

A  log    3.1893180 
t  log    9.6137147 

2.8030327     Nos..  -f  635S.337 

B  log     0.7108786 
fz  log     9.2274294 

9.9383080     Nos..  —      0.867 

C  log     9.0374265 
/3  log    8.8411441 

7.8785706     Nos..  —      0.007 
D  log     8.5910646 
/4  log     8.4548588 

7.0459234     Nos..  -j-      o  .001 


634  .464 


And  equation  6— 
A  ---- 


B  log     0.7108786 
t  log    9.6137147 

2  log        0.3010300 

0.6256233     Nos..     —   4.222 

C  log     9.0374265 
/2  log    9.2274294 

3  log     0.4771213 

8.7419772     Nos..     —    0.055 
D  log     8.5910646 
/3  log     8.8411441 

4  log     0.6020600 


8.0342687     Nos..       +    o.on 
V  = 1542.128 


LONGITUDE.  277 


LXIII. — Longititde  by  Lunar  Culminations — Continued. 
And  equation  (i) — 

I  =  I211 log       4-6354837 

MI  —  fji1  =  os.o64 log        8.8061800 

V  =  i5428.i28 log  ac  6.8118797 


i8-792  . .  .• 0.2535434 

Whence — 

/  =  411  55™  5os  +  is-792  =  411  S5m  5l3-792 

If  m  be  the  observed  increase  of  right  ascension  between  any 
meridian  not  the  first,  (but  of  which  the  longitude  is  well  known,) 
and  the  meridian  sought,  interpolate  the  increase  m,  for  the 
known  meridian  as  well  as  m1  for  that  sought.  Then,  for  m  —  m', 
in  equation  (i),  substitute  ;;/  —  (m1  —  m,),  and  the  result  will  be 
the  corrected  longitude  from  the  first  meridian,  as  before. 

It  often  happens  that  two  observers  do  not  use  the  same 
number  of  wires,  or  do  not  observe  the  same  number  of  stars  at 
the  two  places.  In  such  cases  the  observed  increase  of  the 
right  ascension  of  the  moon's  limb  requires  a  correction,  which 
Mr.  Walker  deduces  as  follows,  from  Gauss's  method : 

For  the  eastern  observatory  and  western  station  respectively, 
let— 

A'  and  A  =  the  observed  AR.  of  a  star ; 
E  =  A'  —  A  for  the  same  star ; 
E'  =  a  similar  value  for  another  star; 
/  and  /'  =  the  number  of  wires  on  which  each  limb  was 

observed ; 
a  and  a'  =  similar  values  for  a  star; 

/ 11 
A  = for  the  moon's  limb ; 

-  for  one  star ; 


a  +  a' 

u'  =  a  similar  value  for  another  star ; 
-T  =  symbol   to   denote   the   aggregate   of  similar 

quantities;  and 
s  =  the  correction  required; 


ASTRONOMY. 


LXIII. — Longitude  by  Lunar  Culminations — Continued, 
then— 


A  u 


/>  +  u 

m  -  m>  +  e 
J.  _  /  J  ~~V~~ 

Also,  calling  W  the  weight  of  each  day's  comparison, 
W  =  7 ^4r-i 


in  which  s  is  the  same  as  as  y  and  <r  =  u  -f-  u'  -f-  «7/  4-  &c. 
For  the  weight  of  the  result  of  all  the  comparisons,  we  have  — 


Let  e  denote  the  probable  error  of  observation,  and  E  the 
probable  error  of  the  final  result;  then, 


(<r  +  A)  z2 

It  also  frequently  happens  that  the  moon  cannot  be  observed  on 
the  middle  wire,  in  which  case  she  is  far  enough  from  the 
meridian  to  have  a  sensible  parallax  in  right  ascension ;  and,  as 
it  may  be  very  desirable  not  to  lose  the  observation,  this  parallax 
must  be  computed  and  applied  to  the  hour-angle  from  the  middle 
wire,  which  is  supposed  to  be  nearly  coincident  with  the  meridian. 

Denoting  this  parallax  in  right  ascension  by  /,  the  horizontal 
parallax  by  w,  the  latitude  of  the  place  of  observation  by  p,  and 
the  true  declination  of  the  moon  by  <?,  we  have  from  the  ordinary 
series  for  the  parallax  in  right  ascension,  neglecting  the  terms 
after  the  first,  which  would  in  this  case  be  insignificant, 

p  =  0  sin  w  cos  <p  sec  d 
in  which  0  is  the  hour-angle,  or  equatorial  interval  in  sidereal 


LONGITUDE.  279 


LXIII. — Longitude  by  Lunar  Culminations — Continued. 

time  from  the  lateral  wire  on  which  the  moon  is  observed  to 
the  central  wire;  so  that,  at  the  instant  of  observation,  the  actual 
distance  of  the  moon's  limb  from  the  central  wire  is — 

0  —  0  sin  w  cos  <p  sec  <5 
and  the  reduction  to  meridian  or  middle  wire  will  be — 

,       0          i  —  sin  w  cos  <p  sec  o 
cos  d  i  —  0.00277;;? 

in  which  ;;/  is  the  motion  of  the  moon  in  right  ascension  in  one 
day,  expressed  in  degrees.  The  upper  sign  is  to  be  used  when 
the  observation  is  on  a  wire  before  and  the  lower  after  the  middle 
wire. 

In  what  precedes  the  approximate  longitude  I1  is  supposed  to 
be  known.     When  this  is  not  the  case,  it  may  be  found  from — 


and  the  interpolation  is  then  to  be  made  for  this  value  of  I'  to 
obtain  the  value  of  m'. 


280  ASTRONOMY. 


LXIV.  —  Longitude  by  the  Electric   Telegraph. 

[From  Cliauvenet's  Practical  Astronomy.] 

It  is  evident  that  the  clocks  at  two  stations,  A  and  B,  may  be 
compared  by  means  of  signals  communicated  through  an  electro- 
telegraphic  wire  which  connects  the  stations.  Suppose  at  a  time 
T  by  the  clock  at  A,  a  signal  is  made  which  is  perceived  at  B  at 
the  time  T'  by  the  clock  at  that  station.  Let  JT  and  JT'  be 
the  clock-corrections  on  the  times  at  these  stations  respectively, 
(both  being  solar  or  both  sidereal.)  Let  x  be  the  time  required 
by  the  electric  current  to  pass  over  the  wire,  then,  A  being  the 
more  easterly  station,  we  have  the  difference  of  longitude  A  by 
the  formula  — 

I  =  (T  +  JT)  -  (T'  +  JT')  +  x  =  *i  +  x 

Since  x  is  unknown  we  must  endeavor  to  eliminate  it.  For 
this  purpose  let  a  signal  be  made  at  B  at  the  clock-time  T",  which 
is  perceived  at  A  at  the  clock-time  Tx//,  then  we  have  — 

A  =  (T'"  +  AT")  -  (T"  +  JT")  -  x  =  A2  -  x 

In  these  formulae  ^  and  J2  denote  the  approximate  values  of 
the  difference  of  longitude,  found  by  signals  east-west  and  west- 
east,  respectively,  when  the  transmission-time  x  is  disregarded, 
and  the  true  value  is  — 


Such  is  the  simple  and  obvious  application  of  the  telegraph  to 
the  determination  of  longitudes  ;  but  the  degree  of  accuracy  of 
the  result  depends  greatly  —  more  than  at  first  appears  —  upon  the 
manner  in  which  the  signals  are  communicated  and  received. 

Suppose  the  observer  at  A  taps  upon  a  signal-key  at  an  exact 
second  by  his  clock,  thereby  producing  an  audible  click  of  the 
armature  of  the  electro-magnet  at  B.  The  observer  at  B  may  not 
only  determine  the  nearest  second  by  his  clock  when  he  hears 
this  click,  but  may  also  estimate  the  fraction  of  a  second  ;  and 
it  would  seem  that  we  ought  in  this  way  to  be  able  to  determine 
a  longitude  within  one-tenth  of  a  second.  But  before  even  this 


LONGITUDE.  281 


LXI V. — Longitude  by  the  Electric  Telegraph — Continued. 

degree  of  accuracy  can  be  secured,  we  have  yet  to  eliminate,  or 
reduce  to  a  minimum,  the  following  sources  of  error  : 

1.  The  personal  error  of  the  observer  who  gives  the  signal; 

2.  The  personal  error  of  the  observer  who  receives  the  signal  and 

estimates  the  fraction  of  a  second  by  the  ear; 

3.  The  small  fraction  of  time  required  to  complete  the  galvanic 

circuit  after  the  finger  touches  the  signal-key ; 

4.  The  armature  time,  or  the  time  required  by  the  armature  at 

the  station  where  the  signal  is  received,  to  move  through 
the  space  in  which  it  plays  and  to  give  the  audible  click ; 

5.  The  errors  of  the  supposed  clock-corrections,  which  involve 

errors  of  observation  and  errors  in  the  right  ascensions  of 
the  stars  employed. 

For  the  means  of  contending  successfully  with  these  sources 
of  error  we  are  indebted  to  our  Coast  'Survey,  which,  under  the 
superintendence  of  Professor  Bache,  not  only  called  into  exist 
ence  the  chronographic  instruments,  but  has  given  us  the  most 
efficient  method  of  using  them.  The  "  Method  of  Star-Signals," 
as  it  is  called,  was  originally  suggested  by  the  distinguished 
astronomer,  Mr.  S.  C.  Walker,  but  its  full  development  in  the 
form  now  employed  by  the  Coast  Survey  is  due  to  Dr.  B.  A. 
Gould. 

Method  of  Star- Signals. 

The  difference  of  longitude  between  the  two  stations  is  merely 
the  time  required  by  a  star  to  pass  from  one  meridian  to  the 
other,  and  this  interval  may  be  measured  by  means  of  a  single 
clock  placed  at  either  station,  but  in  the  main  galvanic  circuit 
extending  from  one  station  to  the  other.  Two  chronographs, 
one  at  each  station,  are  also  in  the  circuit,  and,  when  the  wires 
are  suitably  connected,  the  clock-seconds  are  recorded  upon 
both.  A  good  transit-instrument  is  carefully  mounted  at  each 
station. 

When  the  star  enters  the  field  of  the  transit-instrument  at  A, 
(the  eastern  station,)  the  observer,  by  a  preconcerted  signal  with 
his  signal-key,  gives  notice  to  the  assistants  at  both  A  and  B,  who 
at  once  set  the  chronographs  in  motion,  and  the  clock  then 
records  its  seconds  upon  both.  The  instants  of  the  star's  transits 


282  ASTRONOMY. 


LXIV. — Longitude  by  the  Electric  Telegraph — Continued. 

over  the  several  threads  of  the  reticule  are  also  recorded  upon 
both  chronographs  by  the  taps  of  the  observer  upon  his  signal- 
key.  When  the  star  has  passed  all  the  threads  the  observer  indi 
cates  it  by  another  preconcerted  signal,  the  chronographs  are 
stopped,  and  the  record  is  suitably  marked  with  date,  name  of  the 
star,  and  place  of  observation,  to  be  subsequently  identified  and 
read  off  accurately  by  a  scale.  When  the  star  arrives  at  the  me 
ridian  of  B,  the  transit  is  recorded  in  the  same  manner  upon  both 
chronographs. 

Suitable  observations  having  been  made  by  each  observer  to 
determine  the  errors  of  his  transit-instrument  and  the  rate  of  the 
clock,  let  us  put — 

Tt  =  the  mean  of  the  clock-times  of  the  eastern  transit  of  the 
star  over  all  the  threads,  as  read  from  the  chronograph 
at  A; 

T2  =  the  same,  as  read  from  the  chronograph  at  B; 
T/  =  the  mean  of  the  clock-times  of  the  western  transit  of 
the  star  over  all  the  threads,  as  read  from  the  chrono 
graph  at  A; 

T2'  =  the  same,  as  read  from  the  chronograph  at  B; 
^,  e'  =  the  personal  equations  of  the  observers  at  A  and  B,  re 
spectively; 

T,  -'  =  the  corrections  of  T{  and  T/  (or  of  T2  and  T/)  for  the 
state  of  the  transit-instruments  at  A  and  B,  or  the  re 
spective  "reductions  to  the  meridian;" 
o  T  =  the  correction  for  clock-rate  in  the  interval  T/  —  TI  ; 
x  =  the  transmission-time  of  the  electric  current  between  A 

and  B ;  and 
>•  =  the  difference  of  longitude; 

then  it  is  easily  seen  that  we  have,  from  the  chronographic 
records  at  A, 

;.  =  T/  +  ST  -f  r'  +  c>-  x  -  (T,  +  r  +  e) 

and  from  the  chronographic  records  at  B, 

I  =  IV  +  IT  +  r'  +  e'  +  x  -  (T2  +  r  +  £) 


LONGITUDE.  283 


LXIV.  —  Longitude  by  the  Electric  Telegraph  —  Continued. 
and  the  mean  of  these  values  is 

X  =  [  J  (TV  +  T2')  +  r']  -  [  J  (Ti  +  T2  )  +  r]  +  flT  +  ^  -  <? 
which  we  may  briefly  express  thus: 

A  =  A!  +  ^  -  ^ 
in  which  — 

A!  =  the  approximate  difference  of  longitude  found  by  the 
exchange  of  star-signals,  when  the  personal  equations 
of  the  observers  are  neglected. 

This  equation  would  be  final  if  e?  —  e,  or  the  relative  personal 
equation  of  the  observers,  were  known  ;  however,  if  the  observers 
now  exchange  stations  and  repeat  the  above  process,  we  shall 
have,  provided  the  relative  personal  equation  is  constant, 

X  —  X2  -f  e  —  e1 

in  which  ^2  is  the  approximate  difference  of  longitude  found  as 
before  ;  and  hence  the  final  value  is  — 


I  have  not  here  introduced  any  consideration  of  the  armature- 
time,  because  it  affects  clock-signals  and  star-signals  in  the  same 
manner;  and  therefore  the  time  read  from  the  chronographic 
fillet  or  sheet  is  the  same  as  if  the  armature  acted  instantaneously. 
It  is  necessary,  however,  that  this  time  should  be  constant  from 
the  first  observation  at  the  first  station  to  the  last  observation  at 
the  second,  and  therefore  it  is  important  that  no  changes  should 
be  made  in  the  adjustment  of  the  apparatus  during  the  interval. 

As  the  observer  has  only  to  tap  the  transit  of  the  star  over 
the  threads,  the  latter  may  be  placed  very  close  together.  The 
reticules  prepared  by  Mr.  W.  Wiirdemann  for  the  Coast  Survey 
have  generally  contained  twenty-five  threads,  in  groups  or  "  tallies  " 
of  five,  the  equatorial  intervals  between  the  threads  of  a  group 
being  28.5,  and  those  between  the  groups,  5s;  with  an  additional 
thread  on  each  side  at  the  distance  of  io8,  for  use  in  observations 
by  "eye  and  ear."  Except  when  clouds  intervene  and  render  it 
necessary  to  take  whatever  threads  may  be  available,  only  the 


284  ASTRONOMY. 


LXIV. — Longitude  by  the  Electric  Telegraph — Continued. 

three  middle  tallies,  or  fifteen  threads,  are  used.     The  use  of 
more  has  been  found  to  add  less  to  the  accuracy  of  a  determina- 
•  tion  than  is  lost  in   consequence  of  the   greater  fatigue  from 
concentrating  the  attention  for  nearly  twice  as  long. 

A  large  number  of  stars  may  thus  be  observed  on  the  same 
night ;  and  it  will  be  well  to  record  half  of  them  by  the  clock  at 
one  station  and  the  other  half  by  the  clock  at  the  other  station, 
upon  the  general  principle  of  varying  the  circumstances  under 
which  several  determinations  are  made,  whenever  practicable, 
without  a  sacrifice  of  the  integrity  of  the  method.  For  this 
reason,  also,  the  transit-instrument  should  be  reversed  during  a 
night's  work  at  least  once,  an  equal  number  of  stars  being  ob 
served  in  each  position,  whereby  the  results  will  be  freed  from 
any  undetermined  errors  of  collimation  and  inequality  of  pivots. 
Before  and  after  the  exchange  of  the  star-signals,  each  observer 
should  take  at  least  two  circumpolar  stars  to  determine  the  instru 
mental  constants,  upon  which  r  and  T'  depend.  This  part  of  the 
work  must  be  carried  out  with  the  greatest  precision,  employing 
only  standard  stars,  as  the  errors  of  r  and  r'  come  directly  into 
the  difference  of  longitude.  The  right  ascensions  of  the  "  signal- 
stars"  do  not  enter  into  the  computation,  and  the  result  is,  there 
fore,  wholly  free  from  any  error  in  their  tabular  places ;  hence, 
any  of  the  stars  of  the  larger  catalogues  may  be  used  as  signal- 
stars,  and  it  will  always  be  possible  to  select  a  sufficient  number . 
which  culminate  at  moderate  zenith-distances  at  both  stations, 
(unless  the  difference  of  latitude  is  unusually  great,)  so  that 
instrumental  errors  will  have  the  minimum  effect. 

A  single  night's  work,  however,  is  not  to  be  regarded  as  con 
clusive,  although  a  large  number  of  stars  may  have  been  observed 
and  the  results  appear  very  accordant;  for  experience  shows 
that  there  are  always  errors  which  are  constant,  or  nearly  so,  for 
the  same  night,  and  which  do  not  appear  to  be  represented  in 
the  corrections  computed  and  applied.  Their  existence  is  proved 
when  the  mean  results  of  different  nights  are  compared.  More 
over,  it  is  necessary  to  interchange  the  observers  in  order  to 
eliminate  their  personal  equations.  The  rule  of  the  Coast  Survey 
has  been  that  when  fifty  stars  have  been  exchanged  on  not  less 
than  three  nights,  the  observers  exchange  stations,  and  fifty  stars 


LONGITUDE. 


285 


LXIV. — Longitude  by  the  Electric  Telegraph — Continued. 

are  again  exchanged  on  not  less  than  three  nights.  The  ob 
servers  should  also  meet  and  determine  their  relative  personal 
equation,  if  possible,  before  and  after  each  series,  as  it  may  prove 
that  this  equation  is  not  absolutely  constant. 

Before  entering  upon  a  series  of  star-signals,  each  observer  will 
be  provided  with  a  list  of  the  stars  to  be  employed.  The  prep 
aration  of  this  list  requires  a  knowledge  of  the  approximate 
difference  of  longitude,  in  order  that  the  stars  may  be  so  selected 
that  transits  at  the  two  stations  may  not  occur  simultaneously. 

Example. — For  the  purpose  of  finding  the  difference  of  longi 
tude  between  the  Seaton  station  of  the  United  States  Coast 
Survey  and  Raleigh,  a  list  of  stars  was  prepared,  from  which  I 
extract  the  following  for  illustration.  The  latitudes  are  : 

Seaton  station,  (Washington) <p  =  -f-  38°  537-4 

Raleigh  station,  (North  Carolina) y  =  -f  35    47  .o 

and  Raleigh  is  assumed  to  be  west  from  Washington  6m  30s. 


Seaton      sidereal 

Star. 

Mag. 

a 

6 

time  of  Raleigh 

transit. 

* 

h,  m.     s. 

°        1 

h.     m.      s. 

No.  5036,  B.  A.  C. 

3 

15     09     36 

+  33    52 

15     16    06 

5084 

4-3 

18    58 

37    54 

25     28 

5i3i 

4/2 

27    02 

3i    5i 

33    32 

5*92 

5 

36    35 

26    46 

43    05 

5259 

5 

45    43 

36    07 

52     13 

5322 

4/2 

55    59 

23      12 

16    02    29 

5388 

5 

16    04    09 

45     J9 

10    39 

5463 

3-4 

15    21 

46    40 

21     51 

The  following  table  contains  the  observations  made  on  one  of 
these  stars  at  the  above-named  stations  by  the  United  States 
Coast-Survey  telegraphic  party  in  1853 — April  28 — under  the 
direction  of  Dr.  B.  A.  Gould. 

In  this  table  "  Lamp  W  "  expresses  the  position  of  the  rotation- 
axes  of  the  transit-instruments.  The  first  column  contains  the 
symbols  by  which  the  fifteen  threads  of  the  three  middle  tallies 


286 


ASTRONOMY. 


LXIV. — Longitude  by  the  Electric  Telegraph — Continued. 

were  denoted ;  the  second  column,  the  times  of  transit  of  the  star 
over  each  thread  at  Seaton,  as  read  from  the  chronographs  at 
Seaton ;  the  third  column,  the  times  of  these  transits  as  read 
from  the  chronographs  at  Raleigh ;  the  fourth  column,  the  mean 
of  the  second  and  third  columns ;  the  fifth  column,  the  reduction 
of  each  thread  to  the  mean  of  all,  computed  from  the  known 
equatorial  intervals  of  the  threads ;  the  sixth  column,  the  time 
of  the  star's  transit  over  the  mean  of  the  threads,  being  the 
algebraic  sum  of  the  numbers  in  the  fourth  and  fifth  columns ; 
and  the  remaining  columns,  the  Raleigh  observations  similarly 
recorded  and  reduced. 

Seaton-Raleigh,  1853,  April  28. — Star  No.  5259,  B.  A.  C. 


Thread. 

Seaton  observations,  lamp  W. 

Raleigh  observations,  lamp  W. 

T, 

T2 

Mean 

Red. 

T,  +  T, 

T/ 

T,' 

Mean 

Red. 

T/  +  T./ 

2 

2 

h.  m.   s. 

s. 

s~ 

s. 

J. 

h,    711.      S. 

j. 

j. 

j. 

J. 

Ci 

37-97 

38.00 

37.98 

+  25.49 

3-47 

11.00 

11.00 

+  25.45 

36.45 

C2 

4i-37 

4i-34 

41.36 

22.21 

3-57 

14.58 

14.50 

14-54 

22.25 

36.79 

C3 

44-03 

44.21 

44.12 

19.06 

3-i8 

17.60 

17.55 

17.58 

19.05 

36.63 

C4 

47.81 

47-74 

47.78 

I5-7I 

3-49 

20.88 

20.79 

20.84 

15-85 

36.69 

C5 

50.76 

50.70 

50.73 

12.71 

3-44 

23.90 

23.87 

23.89 

12.70 

36.59 

D! 

56.96 

57-10 

57-03 

6.21 

3-24 

30.19 

30.05 

30.12 

6.32 

36.44 

Da 

0.06 

0.04 

0.05 

3-25 

3-30 

33-34 

33.25 

33-3° 

3.18 

36.48 

D3 

15  46     3.40 

3.38 

3-39 

4-    0.05 

3-44 

i5  S2  36.40 

36.30 

36.35 

+  0.07 

36.42 

D4 

6.70 

6.70 

6.70 

-   3-03 

[3-67] 

39.61 

39-53 

39-57 

-  3-16 

36.41 

D5 

9-58 

9-58 

9-58 

6.28 

3-30 

43.00 

43.00 

43.00 

6.36 

36.64 

EI 

16.03 

15-93 

15.98 

12.54 

3-44 

49.04 

48.81 

48.92 

12.75 

[36.17] 

E2 

19.26 

19.30 

19.28 

15-83 

3-45 

52-30 

52.33 

52-32 

15.90 

36.42 

E3 

22.47 

22.45 

22.46 

18.99 

3-47 

55-50 

55.41 

55.46 

19.10 

36.36 

E4 

'25.60 

25.60 

25.60 

22.23 

3-38 

58.73 

58.60 

58.67 

22.20 

36.47 

E5 

28.60 

28.70 

28.65 

25-33 

3-32 

2.08 

2.08 

2.08 

25-38 

36.70 

Mean  . 

3-392 

Mean  . 

36.535 

The  numbers  in  the  last  column  for  each  station  would  be 
equal  if  the  observations  and  chronographic  apparatus  were  per 
fect;  and  by  carrying  them  out  thus  individually  we  can  estimate 
their  accuracy.  The  numbers  3.67  at  Seaton  and  36.17  at  Ra 
leigh  are  rejected  by  the  application  of  Peirce's  Criterion,  (Method 
of  Least  Squares,)  and  the  given  means  are  found  from  the  re 
maining  numbers. 


LONGITUDE. 


287 


LXIV. — Longitude  by  the  Electric  Telegraph — Continued. 
The  corrections  of  the  transit-instruments  for  this'  star  (d  = 
+  36°  6'.$)  were— 

for  the  Seaton  instrument.,  r  =  —  0.028 
for  the  Raleigh  instrument  T'  =  —  0.193 
The  rate  of  the  clock  was  insensible   in  the  brief  interval 
TV  —  T.     Hence,  neglecting  the  personal  equations  of  the  ob 
servers,  the  difference  of  longitude  is  found  as  follows : 

4  (TV  +  Ta')  +  T'  =  15*   S2m  368-342 
i(Ti   +  T2)  +  r  =  15     46       3-364 


6     32  .978 

In  this  manner  seven  other  stars  were  observed  on  the  same 
night,  and  the  results  were  as  follows  : 


Star. 

fa 

Diff.  from  mean,  =  v. 

m.     s. 

s. 

5036,  B.  A.  C. 

6    33-03 

-f  0.04 

5084,  B.  A.  C. 

33-°9 

-f-   0.  10 

5131,6.  A.  C. 

32.91 

—  0.08 

5192,  B.  A.  C. 

33-00 

-f-  o.oi 

5259,  B.  A.  C. 

32.98 

—  o.  01 

5322,  B.  A.  C. 

33-00 

-f-   O.OI 

5388,  B.  A.  C. 

33-02 

+  0.03 

5463,  B.  A.  C. 

32.91 

—  0.08 

Mean,  fa  =  6    32.  99 

From  the  residuals,  v,  we  deduce  the  mean  error  of  a  single  de 
termination  by  one  star, 


and  hence  the  mean  error  of  the  value  6m  32^99  is  — 

,    os.o6 

£0    =    ±  —j=   =    ±   Os.02 

But  this  error  will  be  somewhat  increased  by  those  errors  of 
the  instruments  which  are  constant  for  the  night,  and  not  repre 
sented  in  T  and  r7,  and  by  the  errors  of  the  personal  equations 
yet  to  be  applied.  Moreover,  a  greater  number  of  determina 
tions  should  be  compared  in  order  to  arrive  at  a  just  evaluation 
of  the  mean  error. 


288  ASTRONOMY. 


LXV. — Formula  for  Probable  Error  and  Precision. 
[Contributed  by  Lieutenant  Mercur,  Corps  of  Engineers.] 

i.  Let — 

;;/  =  the  number  of  observations; 
;/,  n',  &c.  =  results  found  by  observation ; 

x  =  their  arithmetical  mean ; 

i',  z>',  &c.  =  (#—;/),  (x — ;/'),  &:c.  =  the  residual  errors  of  ob 
servation; 

e  =  the  mean  error  of  ;/,  n',  &c. ; 
7]  =  the  mean  of  errors,  (arithmetical;) 
E0  =  the  mean  error  of  final  result; 
E!  =  the  mean  error  of  the  observation  assumed  as 

the  standard  of  excellence; 

r  =  the  probable  error  of  a  single  observation, ;/,  n',  &c. ; 
R  =  the  probable  error  of  the  final  result,  x; 
h  =  the  measure  of  exactness  of  a  single  observation, 

X  «',  &c.; 

H  =  the  measure  of  exactness  of  the  final  result,  x ; 
/,  /',  £c.  =  the  weights  of  different  observations  or  sets  of 

observations; 
ef,  e",  &c.  =  the  mean  errors  of  observations  corresponding 

to/,/,  &c.; 
I  =  symbol  representing  the  sum; 


r  —  0.6745  e  =  0.8453  >j  =  0,8453  ~: 


Iv 


(m  —  i) 


/-        T-  /•  /       2v*  /°-4S49S^7;2  r 

R  =  0.6741;  E0  —  0.6745     /  -  r-  =     /  —  Y  .     =  —  = 

9\m(m*-i)       V   m  (m  —  i)          -/  m 

7,^0.46936  H  =  //V^ 

r 

2.  By  "the  weight  of  any  determination"  is  meant  its  relative 
approximation  to  the  true  value. 

It  may  be  measured  by  the  number  of  observations  (each  of 
which  may  be  considered  as  good  as  the  other,  and  of  which  one 
is  assumed  to  represent  the  unit  of  excellence)  necessary  to  give 
a  result  equally  near  the  true  value. 


ERROR   AND    PRECISION.  289 


LXV.  —  Formula  for  Probable  Error,  &c.  —  Continued. 

Then,  since  the  weights  of  observations  are  inversely  as  the 
squares  of  their  mean  or  probable  errors, 


and  when  we  arbitrarily  assign  'weights  to  each  observation  or 
set  of  observations, 


-i) 

in  which  ;//  =  the  number  of  observations  or  sets  of  observations 
whose  weights  are/,/',  &c. 

When  observations  are  combined  by  weights,  the  probable  error 
is  given  by  the  formula — 


R  =  0.6741; 

745 


3.  PEIRCE'S  Criterion  for  the  rejection  of  doubtful  observations. 
To  apply  this  to   any  set  of   observations  involving  but  one 
unknown  quantity :  * 

Let— 

m  =  the  number  of  observations  taken; 
;/  =  the  number  of  doubtful  observations  to  be  re 
jected,  (to  be  found  by  trial;) 
s  =  the  mean  error  of  one  observation  in  the  set  of 

///  observations; 
i',  v'j  &c.  =  the  residual  errors  of  the  observations;  and 

-/.  =  the  ratio  of  the  required  limit  of  error  for  the  re 
jection  of  n  observations  to  the  mean  error  -, 
so  that  xs  is  the  limiting  error. 

Find  from  the  table  the  value  of  x2  for  n  —  i,  n  =  2,  n  =  3, 
&c.,  in  succession,  and  reject  all  observations  in  which  xe>z'; 
stopping,  however,  when  the  value  of  xs  found  for  any  particular 
value  of  n  does  not  reject  any  observations  (not  already  rejected) 
for  a  value  of  n  numerically  one  less. 

*  For  rules  for  determining  mean  and  probable  errors,  and  for  applying  the 
Criterion  to  cases  involving  more  than  one  unknown  quantity,  see  Chauvenet's 
Manual  of  Spherical  and  Practical  Astronomy,  vol.  II,  pages  469  to  566. 


290 


ASTRONOMY. 


LXV. — Formula  for  Probable  Error,  &c. — Continued. 

Example. — To  determine  the  value  of  one  turn  of  the  microm 
eter  of  a  zenith-telescope,  the  following  "reduced  intervals"  were 
obtained,  each  corresponding  to  the  ten  turns  of  the  micrometer : 


Between 
observa 

Reduced 
intervals. 

V 

tions. 

111.      S. 

i  and  10 

20    35.1 

II  .0 

121.  OO 

2  and  12 
3  and  13 

43-9 
41.2 

2.2 

4-9 

4.84 
24.01 

when  n  —  i             K*  —      3.707 
K*  ea  =  204.148 

4  and  14 

51-2 

5-i 

26.OI 

K  e     =     14.28 

5  and  15 

46.6 

°-5 

00.25 

which  rejects  (n  and  21) 

6  and  16 

44.0 

2.  I 

4.41 

7  and  17 

38.2 

7-9 

62.41 

in  ~  1  1 

8  and  18 

54-2 

8.1 

65.61 

when  n  =  2            K*  —     2.621 

9  and  19 

47-9 

1.8 

3-24 

K*  e2  ==  144.341 

10  and  20 

43-9 

2.2 

4.84 

which  rejects  none  other. 

ii  and  21 

21    OI  .4 

15-3 

234.09 

Sum  =  228  27.6 

£  v'l---=  550.71 

x  =    20  46.1 

*l=    55-Q71 

After 

rejecting  the  interval  between  (n  and  21)  the  probable 

error  is 

found  as  follows  : 

Between 
observa 
tions. 

Reduced 
intervals. 

V 

V* 

•/;/.    j. 

i  and  ii 

20    35.1 

9-5 

90.25 

2  and  12 

43-9 

0.7 

•49 

3  and  13 

41.2 

3-4 

11.56 

4  and  14 

51.2 

6.6 

43-56 

5  and  15 

46.6 

2.0 

4.00 

6  and  16 

44.0 

0.6 

.36 

7  and  17 

38.2 

6.4 

40.96 

8  and  18 

54-2 

9.6 

92.  16 

9  and  19 

47-9 

3-3 

10.89 

10  and  20 

43-9 

0.7 

•49 

Sum  = 

207  26.2 

,..= 

294.72 

X 

20    44.62 

€*  = 

32-749 

/  294  -.72    _       , 


9°-25  |  pro5abie  error  of  single  set    =  0.6745  x  5.72  or 
r  =  ±  3  s  -86 


Probable  error  of  final  result  =  '    or 


This  is  for  ten  revolutions  ;  for  one  revolution 
the  probable  error  is  ±  o«  .122. 


PEIRCES    CRITERION. 

Peirce's  Criterion. 
Values  of  K"  for  //  =  i. 


291 


n 

i 

2 

3 

4 

5 

6 

7 

8 

9 

1.480 

j 

4 

1.912 

I.l63 

5 

2.278 

!-439 

6 

2.592 

1.687 

i.  208 

7 

2.866 

i  .910 

1.409 

:        I-°45    ' 

8 

3.109 

2.  112 

1.589 

1.229 

9 

3.327 

2.295 

L753 

i        1.388            I.OQI 

IO 

3-526 

2.464 

1.904 

1.242 

ii 

3-707 

2.621 

2.045 

;        1.662 

J-373 

I.  122 

12 

3.875 

2.766 

2.176 

i        1.785 

1.492 

1.249 

1.018 

13 

4.029 

2.902 

2.299 

;     1.901 

i  .604 

1.362 

I.I45 

14 

4-173 

3.030 

2.416 

!     2.009 

1.709   |     1.465 

1-255 

1     1.053 

15 

4-309 

3-  I5I 

2.526 

2.  Ill 

1.807  I     Io6i 

J-354 

1.163 

16 

4-436 

•  3-264 

2.630 

2.207 

1.898   ;      1.651 

•445 

1-259 

1.080 

J7 

4*  555 

3-371 

2.729 

;    2.300      1.985  |    1.736 

•  529 

1-347 

i  .  176 

18 

4  •  668 

3-475 

2.824 

2.389      2.069  i    1.817 

.609 

.     1.428 

1.261 

J9 

4.776 

3-57T 

2.914 

2.474     2.150  •    1.895 

.685 

i  504 

r-341 

20 

4.878 

3.664 

3.001 

:       2.556           2.227    :       J-970 

•  757 

i     1-576 

21 

4.9/5 

3-755 

3.084 

1       2.634 

2.301    !     2.041 

.827 

i     1.644 

1.483 

22 

5.068 

3.840 

3.164 

;   2.709     2.373  !   2.109 

.893 

1.710 

1  •  549 

23 

5-  157 

3.923 

3.240 

;     2.782       2.442       2.176 

•  957 

T-773 

i  .612 

24 

5-242 

4.002 

3.3I5 

2.852 

2.509   !     2.240 

.019 

1-833 

1.671 

25 

5-324 

4.078 

3-387 

;      2.920        2.573   •     2.302 

.079 

1.892 

1.729 

26 

5-403 

4-  I5I 

3.456 

2.986           2.636    :       2.362 

•T37 

1.948 

1.784 

27 

28 

5-479 

4.222 

•^s88 

:       3.049           2.697    i      2.420 

.194 

2.003 

1.838 

29 

5.622 

4.291 

4-358 

3-651 

3.111           2.756    ;       2.477 

3.171  ;    2.813  ;    2.532 

.249 
.302 

2.056 
2.108 

1.941 

30 

5-690 

4.422 

3-712 

3.229     2.869  ;    2.586 

•354 

2.158 

1.990 

31 

5  •  756 

4-484 

3-772 

3-285 

2.923        2.638 

.404 

2.207 

2.038 

32 

5.820 

4-545 

3.829 

3-34° 

2.976        2.689 

•454 

2-255 

2.085 

33 

5.882 

4.604 

3-884 

3-394 

3.028        2.738 

.502 

2.302 

2.130 

34 

5-942 

4.661 

3-939 

3-446 

3.078        2.787 

-549 

2-347 

2-175 

35 

6.001 

4-7I7 

3-992 

3-497 

3.127   f    2.834 

•594 

2.392 

2.218 

36 

6.058 

4-77* 

4.044 

3-547 

3-174 

2.880 

•639 

2.436 

2.261 

37 
38 

6.  113 
6.167 

4^874 

4-095 
4.144 

3-595 
3-643 

3.221 

3.267 

2.926 
2.970 

.683 
.726 

2.478 
2.520 

2.302 
2-343 

39 

6.2IQ 

4-925 

4.192 

3.689 

3-312 

3.013 

.768 

2.561 

2-383 

40 

6.270 

4-974 

4-239 

3-734 

3-055 

.809 

2.601 

2.422 

41 
42 

6.320 
6.360 

5.022 
5.069 

4.285 

3-779 
3.822 

3.398 
3-440 

3-°97 
3.138  . 

.849 
.888 

2.640 
2.678 

2  .  460 

2-497 

43 

6.416 

5.H4 

4-375 

3-865 

3.481 

3-178  i 

•927 

2.  716 

2-534 

44 
45 

6.463 

6.508 

5-159 
5.202 

4.418 
4.460 

3.906        3.521 
3.947        3o6i 

3-217  ; 
3-255 

-965 
3.002 

2-753 
2.789 

2.570 
2.606 

46 
47 
48 
49 

6-552 
6.596 
6.639 
6.681 

•     5-245 
5-287 
5-328 
5.368 

4.501 
4-542 
4.581 
4.620 

3-987 
4.026 
4-065 
4.103 

3.600 
3-638 
3-675 
3-712 

3-293 
3.33°  1 
3-366  j 
3.401 

3-°39 
3-075 
3.110 
3-I45 

2.825 
2.860 
2.894 
2.928 

2.641 
2.675 
2.708 
2.741 

50 

6.720 

5-408 

4-657 

4.140 

3.748 

3-436 

3-1.79 

2.962 

2-774 

5i 
52 
53 
54 
55 
56 
57 

6.761 
6.800 
6.838 
6.876 
6.913 
6.950 
6.986 

5-447 
5-484 
5-522 
5-559 
5-595 
5.630 
5-665 

4-695 
4-732 
4.768 
4.804 
4-839 
4.873 
4.907 

4.176 
4.212 
4.247 
4.282 
4.316 
4-349 
4-382 

3-784 
3.819 

iisj 

3.920 
3.952 
3.984 

3-471 
3-505  i 
3-538  i 
3-571  ! 
3-603  ! 
3-635  | 

3-213 
3.246 
3-279 
3-311 
3-342 
3-373 
3-4°4 

2-994 
3.027 

3-059 
3.090 
3.121 

3^181 

2.806 
2.838 
2.869 
2.899 
2.929 

2-959 
2.988 

58 

7.021 

5-699 

4.941 

4-4*5 

4.016     3.697  1 

3-434 

3.210 

3-OI7 

59 
60 

7.056 
7.090 

5-733 

5.766 

4-974 
5.006 

4-447 
4.478 

4.047     3.728 

4.078        3.758  : 

3-463 
3-492 

3-239 

3.268 

3-046 
3-074 

Geographical  Positions. 


Latitude. 


Cambridge,  Observatory 42  22  48.  I 

Quebec,  Citadel 46  48  1 7.  3 

New  York,  Observatory 40  43  48.  5 

Oswego,  Court-House 43  27  49.1 

WASHINGTON,  Observatory 38  53  38.  8 

Buffalo,  Michigan  and  Exchange  streets 42  52  41.8 

Detroit,  New  L.  S.  Observatory 42  19  58.  6 

Chicago,  City  Hall 41  53  06.  2 

Saint  Louis,  Washington  University 38  37 

Saint  Paul,  Custom-House 44  53 

Fort  Leaven  worth,  Engineer  Observatory 39  21 

Omaha,  Coast- Survey  Observatory 41   16 

Denver,  Mint 39  45  01.8 

Salt  Lake,  Coast-Survey  Observatory 40  46 

San  Francisco,  Washington  Square 374755-3 


Longitude 
west       from 
Greenwich. 
Ji.  in.    s. 
4  44  31.04 
4  44  49-42 

4  55  56-  65 

5  06  01. 05 

5    08    12.  12 

5  15  27.58 
5  32  12-24 

5  50  32. 08 

6  oo  49. 02 

6    12   21. 84 

6  19  39-35 
6  23  46. 33 

6  59  58-  72 

7  27  35-45 

8  09  38.23 


TABLES    AND     FORMULAE 


PAR  T     I  V. 


APPENDIX. 


APPE  N  DIX. 


LXVI. — Field  Magnetic  Observations. 
[By  Captain  CHAS.  W.  RAYMOND,  Corps  of  Engineers.] 

These  observations  have  for  their  object  the  determination  of 
the  magnetic  declination,  dip,  and  intensity,  at  any  given  time 
and  place. 

The  following  instructions  have  reference  to  the  determination 
of  the  magnetic  elements  on  land.  It  will  be  supposed  that  the 
theodolite  magnetometer  and  dip- circle  are  the  instruments  em 
ployed  ;  the  first  to  determine  the  declination  and  the  horizontal 
component  of  the  intensity,  and  the  second  to  determine  the  in 
clination  or  dip. 

Magnetic  Declination, 

The  magnetometer  having  been  mounted  upon  its  tripod,  or 
upon  a  sound  post  firmly  imbedded  in  the  ground,  the  horizontal 
limb  and  the  rotation-axis  of  the  telescope  must  be  leveled,  the 
vertical  wire  of  the  telescope  made  truly  vertical,  and  its  collima- 
tion-error  reduced.  The  magnet  must  then  be  suspended  by  as 
few  filaments  as  possible  ;  four  or  five  are  usually  required.  The 
magnet  is  then  made  horizontal  by  adjusting  the  balancing-ring, 
the  position  of  which  should  be  carefully  preserved  throughout 
the  experiments.  In  order  to  adjust  the  magnet-scale  to  the 
stellar  focus  of  its  lens,  the  telescope  must  be  turned  upon  the 
sun  or  a  star,  and  adjusted  to  perfectly  distinct  vision.  The  tele 
scope  must  then  be  turned  upon  the  suspended  magnet,  and  the 
scale-ring  screwed  in  or  out  until  the  scale  is  seen  with  perfect 
distinctness. 

The  lines  of  detorsion  and  collimation  must  now  be  brought 
into  the  plane  of  the  magnetic  meridian  by  the  following  method : 
The  magnet  being  suspended,  turn  the  instrument  in  azimuth 
until  the  scale  is  seen  through  the  telescope.  Remove  the 
magnet  and  suspend  the  brass  detorsion-cylinder.  Bring  the 
axis  of  the  cylinder,  by  estimation,  into  the  plane  of  the  magnetic 
meridian  by  turning  the  torsion-circle.  Remove  the  cylinder 
and  suspend  the  declination-magnet.  Turn  the  instrument  in 
azimuth  until  the  vertical  wire  of  the  telescope  bisects  the  scale- 
zero.  Remove  the  declination  and  suspend  the  short  magnet. 


296  APPENDIX. 


LXVL — Field  Magnetic  Observations,  &c. — Continued. 

Turn  the  torsion-circle  until  the  vertical  wire  of  the  telescope 
coincides  with  the  scale-zero.  Exchange  the  magnets  and  adjust 
as  before.  The  time,  temperature,  and  readings  of  the  verniers 
and  torsion-circle  should  be  recorded.  This  method  requires 
that  the  magnets  and  detorsion-cylinder  should  be  of  equal  weight. 
With  the  cylinder  alone,  the  plane  of  detorsion  may  be  determined 
to  within  about  one  degree,  an  error  which  does  not  seriously 
affect  the  accuracy  of  declinations  observed  in  ordinary  field- 
work. 

The  instrument  is  now  in  adjustment  for  observations  of  dec 
lination. 

The  angular  value  of  one  scale-interval  and  the  scale-zero  of 
each  magnet  employed  must  be  determined  at  some  convenient 
time.  The  former  is  unchangeable.  The  latter  .must  be  occa 
sionally  redetermined,  as  it  is  liable  to  change  through  accident. 
To  determine  the  angular  scale-value,  fix  the  magnet  in  the 
position  which  it  occupies  when 'suspended,  in  such  a  way  that 
the  instrument  may  be  moved  in  azimuth  without  disturbing  it. 
Turn  the  instrument  in  azimuth  until  the  vertical  wire  of  the 
telescope  coincides  with  a  scale-division.  Record  the  vernier 
and  scale  readings.  Turn  the  instrument  until  the  vertical  wire 
coincides  with  another  division,  and  record  as  before.  Repeat 
over  different  parts  of  the  scale  until  a  sufficient  number  of  ob 
servations  have  been  obtained.  Each  pair  of  observations 
furnishes  a  single  determination  of  the  required  value.  The 
probable  error  of  the  mean  value  may  be  determined  by  the 
method  of  least  squares. 

To  determine  the  scale-zero,  or  reading  of  the  magnetic  axis, 
suspend  the  magnet,  turn  the  instrument  in  azimuth  until  the 
vertical  wire  of  the  telescope  coincides  with  a  division  near  the 
middle  of  the  scale,  and  record  the  scale-reading.  Invert  the 
magnet,  and  record  the  reading  corresponding  to  this  position. 
Move  the  instrument  slightly  in  azimuth,  and  repeat  this  operation 
until  three  or  four  readings  with  the  scale  erect,  and  as  many 
with  the  scale  inverted,  have  been  obtained.  The  scale-zero  may 
then  be  determined  by  the  method  of  alternate  means,  (see  Form 
A.)  These  observations  should,  if  practicable,  be  made  about 
the  epoch  of  the  day  when  the  magnet  is  stationary. 


FIELD    MAGNETIC    OBSERVATIONS.  297 


LXVI. — Field  Magnetic  Obsewations,  <5rv. — Continued. 

The  co-efficient  of  torsion  may  be  determined  as  follows :  The 
declination-magnet  being  suspended  and  the  instrument  in  ad 
justment  for  observations  of  declination,  record  the  readings  of 
the  scale  and  torsion-circle.  Turn  the  torsion-circle  through 
an  angle  of  90°,  and  record  this  difference  of  arc  and  the  corre 
sponding  scale-reading.  Turn  the  torsion-circle  180°  in  the 
reverse  direction,  and  record  as  before.  Finally,  turn  the  torsion- 
circle  back  to  its  original  position,  and  repeat  the  readings  and 
record. 

Computation. 

H  u__ 

F   '"  90°  —  u 

TT 

i  +  -^  =  co-efficient  of  torsion;  and 

u  =  difference  in  scale-readings  (reduced  to  arc)  cor 
responding  to  a  change  of  direction  of  the 
magnet  caused  by  twisting  the  suspension- 
thread  through  an  angle  of  90°. 

The  mean  value  of  u  deduced  from  the  observations  is  em 
ployed.  For  convenience  the  co-efficient  of  torsion  is  usually 
applied  to  the  angular  value  [a]  of  the  scale-interval,  (see 
Form  B.) 

At  some  convenient  time,  while  the  instrument  is  in  position, 
the  vernier-readings  corresponding  to  the  astronomical  meridian 
must  be  determined,  either  by  turning  the  telescope  upon  a  point 
of  which  the  azimuth  is  known,  or  directly,  by  any  suitable 
method.  * 

The  preliminary  adjustments  and  determinations  having  been 
made,  the  instrument  is  left  in  position  for  observations  of  the 
diurnal  variations  in  declination.  At  some  time  early  in  the 
morning,  the  north  end  of  the  magnet  attains  its  most  easterly 
position,  which  is  called  the  morning  eastern  elongation.  The 
record  of  scale-readings  must  be  commenced  early  enough  to 
include  this  elongation.  When  this  point  is  fairly  passed,  or  the 
north  end  of  the  magnet  has  fully  commenced  its  westerly 
motion,  the  readings  may  be  discontinued  until  about  noon,  when 
they  must  be  resumed  and  continued  until  the  western  elongation 
has  been  observed,  and  the  easterly  motion  has  fairly  set  in. 


298 


APPENDIX. 


LXVI. — Field  Magnetic  Observations,  &c. — Continued. 

The  telescope  should  be  reversed  at  each  observation,  and  a 
mean  of  the  readings  in  the  two  positions  should  be  taken. 
The  temperature  should  be  noted.  The  readings  should  be  made 
half  or  quarter  hourly  during  the  periods  of  observation.  The 
observations  of  the  first  day  will  determine  the  approximate 
times  of  elongation,  or  turning-hours,  in  accordance  with  which 
the  periods  of  observation  are  to  be  subsequently  regulated. 

Computation. 

/  H  \ 

*  =  *±'(*   +p  )(<-*+» 

3  ==  value  of  declination  for  the  day ; 
o'  =  declination  at  instant  of  final  'instrumental   ad 
justment,  which   is  -J        1  when  the  magnetic 

meridian  is  {  we^    1  of  the  true  north  meridian  ; 
(  east  ) 

a  =  angular  scale-value ; 

TT 

i  +  u  =  co-efficient  of  torsion ; 

e  =  mean  of 'scale-readings  at  the  elongations,  which 


s  =  scale-reading  of  magnetic  axis,  (zero  of  magnet- 
scale,)  which  is  {  +  |  when  {  jf^^  }  than  ,; 


^ 

r  =  difference  between  scale-readings  at  elongations, 
or  daily  range. 


j  =  factor  tor  reduction  to  the  mean  ol  no 
vations.     It  may  be  taken,  with  its 
the  following  table  : 

7 

urly  obser- 
sign,  from 

i 

January.  
February  .  .  . 
March 

—  o.  089 
—  0.040 

—  O  OIQ 

May  . 

—  o.  01^ 

j  September  . 
i  October 

-  o.  044 
o  006 

•  " 

June 

—  (••   O   OIO 

Tulv 

+  o  ooc 

November  . 
'  December.. 

.  .  —  o.  096 

O    I  ^4. 

April 

—  o.  068 

j  uv  

August  .. 

..  —  0.023 

• 

1 

FIELD    MAGNETIC    OBSERVATIONS.  299 


LXVI. — Field  Magnetic  Observations,  crv. — Continued. 

The  correction  to  d*  is  positive  or  negative  according  as  it  in 
dicates  a  motion  from  or  toward  the  true  meridian.  (See  Form  C. ) 

Magnetic  Intensify. 

For  the  determination  of  the  horizontal  intensity  two  distinct 
series  of  experiments  are  required — experiments  of  deflection  and 
experiments  of  oscillation. 

The  experiments  of  deflection  are  made  as  follows :  The  de 
flection-bar  is  made  fast  in  its  position,  and  the  copper  damper 
placed  within  the  box.  The  instrument  is  then  adjusted  as  for 
observations  of  declination,  the  short  magnet  being  suspended. 

The  experiments  should  be  made,  when  practicable,  at  three 
distances.  The  first  position  of  the  deflector  should  be  at  about 
three  times  its  length  from  the  suspended  magnet ;  the  third,  at 
a  distance  about  one-third  greater ;  and  the  second  midway  be 
tween  the  other  two.  These  distances  are  measured  from  center 
to  center.  At  the  beginning,  the  verniers  are  read  and  time 
noted,  in  order  to  follow  changes  of  declination.  The  tempera 
ture  is  recorded  at  each  observation.  The  long  magnet  is  placed 
upon  its  carriage  on  the  deflection-bar,  on  either  side  of  the  sus 
pended  magnet,  and  at  the  nearest  distance.  The  instrument  is 
then  turned  in  azimuth  until  the  vertical  wire  of  the  telescope 
coincides  with  the  scale-zero.  The  time  is  then  noted  and  the 
verniers  read.  The  carriage  is  then  moved  to  the  next  distance, 
and  finally  to  the  greatest  distance,  and  the  observation  is  re 
peated  at  each  position. 

The  deflector  is  then  reversed  on  its  carriage,  and  the  observa 
tions  are  repeated  at  the  three  distances,  beginning  with  the 
greatest  and  ending  with  the  least.  At  the  nearest  distance  the 
magnet  is  again  reversed,  and  the  complete  set  of  observations  is 
repeated,  in  order  to  obtain  a  double  set  of  results.  The  deflector 
is  now  placed  on  the  opposite  side  of  the  suspended  magnet  at 
the  nearest  distance.  A  double  set  of  experiments  similar  to 
that  already  described  is  then  made. 

At  suitable  intervals  special  observations  should  be  made  to 
measure  changes  of  declination.  For  this  purpose  the  deflector 
is  removed,  and  the  instrument  is  turned  in  azimuth  until  the 
vertical  wire  of  the  telescope  coincides  with  the  scale-zero.  The 


3°°  APPENDIX. 


LXVI. — Field  Magnetic  Observations,  &c. — Continued. 

verniers  are  then  read  and  the  temperature  and  time  noted. 
From  these  observations  we  may,  by  simple  interpolation,  deter 
mine  with  sufficient  accuracy  corrections  for  the  reduction  of  the 
observed  angles  of  deflection  to  the  same  declination. 


Computation. 


m 


m  =  magnetic  moment  of  the  deflector  ; 
X  =  horizontal  intensity  ; 

;•  =  distance  between  the  centers  of  the  magnets  ;  and 
P  =  constant  depending  on  the  distribution  of  magnetism 
in  the  magnets. 

1  he  value  of  -^  must  be  computed  for  each  distance  separately 
and  a  mean  adopted.  (See  Form  D.) 

The  correction  depending  upon  the  constant  P  may  be  neg 
lected  when  there  is  a  considerable  difference  between  the  lengths 
of  the  magnets.  Its  value  is  greatest  when  the  magnets  are 
equal  in  length,  and  zero  when  the  lengths  are  in  the  proportion 
of  i  to  1.224. 

To  determine  the  value  of  P,  deflections  are  made  alternately 
at  two  different  distances,  which  should  be  in  the  proportion  of 
i  to  1.32.  About  twenty-five  corresponding  sets  should  be 
obtained. 

Computation. 
A  —  A, 


P  = 
A 


.  ?' 

'     >  =  value  of  -Y  for 

*M  j  -A.  ^  a.^»iigv^i     j  {  '  1 

To  reduce  the  angles  of  deflection  determined  from  different 
sets  to  the  same  temperature  : 

sin  7/0 
sin  «  =  -—  (4  _-{]q 

7/0  =  observed  angle  at  temperature  /0 ; 
u   =  angle  reduced  to  standard  temperature  /;  and 
q   =  temperature-constant,  determined  as  explained  here 
after. 


FIELD    MAGNETIC    OBSERVATIONS.  301 


LXVI. — Field  Magnetic  Observations,   erv. — Continued. 

The  experiments  of  oscillation  are  made  as  follows :  The  in 
strument  having  been  adjusted  as  for  observations  of  declination, 
with  the  long  magnet  suspended,  and  the  co-efficient  of  torsion 
having  been  determined,  the  magnet  is  made  to  oscillate  hori 
zontally  by  attracting  or  repelling  one  of  its  poles.  The  impulse 
should  be  sufficient  to  make  it  oscillate  beyond  the  limits  of  the 
scale  for  at  least  ten  minutes,  as  steadiness  of  motion  is  thus 
acquired.  All  vertical  oscillations  must  be  carefully  checked. 
When  the  amplitude  is  sufficiently  reduced,  the  scale  is  read  at 
the  limits  of  an  oscillation,  and  a  division  near  the  mean  of  these 
readings  is  selected  as  the  zero  or  point  at  the  passage  of  which 
the  times  are  to  be  noted.  The  intervals  of  oscillation  may  now 
be  noted  in  the  following  way :  The  approximate  interval  corre 
sponding  to  six  oscillations  is  noted  for  convenience.  The  instant 
of  passage  is  then  noted  at  every  sixth  oscillation  up  to  the 
sixtieth;  then  at  the  hundredth,  two  hundreth,  and  three 
hundreth ;  then  at  every  sixth  oscillation  up  to  the  three  hundred 
and  sixtieth ;  and  then  at  the  four  hundredth,  if  it  be  desirable  to 
prolong  the  observations  to  this  extent.  The  number  of  oscilla 
tions  timed  should  depend  on  the  length  of  the  interval.  The 
entire  time  of  observation  should  not  exceed  a  quarter  of  an 
hour.  The  approximate  time  of  ten  oscillations  is  computed  for 
convenience  at  the  sixtieth.  For  the  semi-oscillations  timed  the 
magnet  will  always  move  in  the  same  direction.  The  amplitude 
of  oscillation  at  the  beginning  and  end  of  the  experiments  should 
be  noted.  At  suitable  intervals  the  mean  reading  of  the  scale 
should  be  observed,  since  the  zero  is  liable  to  changes  due  to 
variations  of  declination. 

The  temperature  should  be  observed  at  intervals,  the  bulb  of 
the  thermometer  being  placed  within  the  box. 

Computation. 

*2K 
mX  =  -7p- 

;;/  =  magnetic  moment  of  the  magnet; 
X  =  horizontal  intensity ; 

*  =  3-M-I59; 
K  =  moment  of  inertia  of  magnet  -and  stirrup ; 


302  APPENDIX. 


LXVI.  —  Field  Magnetic  Observations,  6-V.  —  Continued. 
T  =  corrected  time  of  oscillation  ; 

T*  ^  T*(.  +  5)  (i  -('-') 

T'  =  observed  time  of  oscillation  ; 
;}  =  te»peratnre  of  magnet  when 

tj  —  temperature-constant,  or  change  in  magnetic  moment 
for  a  change  in  temperature  of  i°  Fahrenheit. 
(See  Form  E.) 

To  determine  the  moment  of  inertia  of  the  magnet  and  stirrup, 
[K],  the  moment  of  inertia  of  the  inertia-ring  must  first  be  com 
puted.  For  this  purpose  accurate  measurements  of  its  outer  and 
inner  radii  (in  decimals  of  a  foot)  and  the  weight  of  the  ring  (in 
grains)  are  required.  These  data  are  usually  furnished  by  the 
maker. 

The  instrument  being  in  adjustment  with  the  long  magnet 
suspended,  the  inertia-ring  is  balanced  upon  the  magnet  by 
means  of  the  balancing-blocks.  The  time  of  a  single  oscillation 
is  then  determined  as  before  described.  The  load  is  then  removed 
and  the  time  of  oscillation  is  again  determined.  At  least  twelve 
sets  of  these  experiments  should  be  made.  A  separate  determi 
nation  of  the  co-efficient  of  torsion  must  be  made  for  the  loaded 
magnet.  The  temperature  must  be  recorded  throughout  the 
experiments. 

Computation. 


K"  =  4  (r*  +  r'2)  w 

K  =  moment  of  inertia  of  the  suspended  mass  ; 
K"  =  moment  of  inertia  of  the  ring  ; 

T'  )  ,    •         r    •     i         -11  s        r  (  loaded      ) 

T  J  =  corrected  time  of  single  oscillation  of  |  unloaded  f 

magnet  ; 


radius  of  rinS>  in  feet  ;  and 


ft'  =  weight  of  ring,  in  grains. 


FIELD    MAGNETIC    OBSERVATIONS.  303 


LXVI. — Field  Magnetic  Observations,  crV. — Continued. 

The  values  of  -2  K  for  different  temperatures  should  be  tabu 
lated.  The  co -efficient  of  expansion  for  brass  (o.ooooi)  may  be 
employed  for  their  computation. 

The  change  in  magnetic  moment  for  a  change  in  temperature  of  i° 
Fahrenheit,  \q\,  is  best  determined  by  the  method  of  deflections. 
The  magnet  for  which  q  is  to  be  determined  is  the  deflector.  At 
least  three  consecutive  sets  of  deflections  should  be  made,  the 
first  and  third  being  at  about  the  same  temperature,  and  the 
intermediate  set  at  a  very  different  one.  A  mean  of  the  results 
from  the  first  and  third  sets  must  be  compared  with  the  result 
from  the  intermediate  set.  The  required  difference  of  tempera 
ture  may  be  produced  by  a  jacket  of  ice  and  hot  water,  or 
advantage  may  be  taken  of  extreme  natural  temperatures. 

Computation, 
a  11  cot  // 


q  =  temperature-constant ; 

/  \        (  higher  ) 

/!,}  =  !  lower  }  temperature ; 

//  =  difference  of  scale-readings  corresponding  to  /  —  /0; 
a  =  angular  value  of  one  scale-interval ;  and 
;/  =  angle  of  deflection  corresponding  to  /„. 

Computation  of  the  Horizontal  Intensity  and  Magnetic  Moment. 


X=     /« 

V  a 


X  :=  absolute  horizontal  intensity; 

;//..  =  magnetic  moment  of  deflecting  and  oscillating  magnet ; 
«  =  ;;/X,  determined  by  experiments  of  oscillation ;  and 

•t  =  -^ - ,  determined  by  experiments  of  deflection. 

Strictly,  the  values  of  a  and  /?  should  be  corrected  for  the  effect 
of  induction.     These  corrections    are  very  small,   and   require 


3°4  APPENDIX. 

LXVI. — Field  Magnetic  Observations,  &c. — Continued. 

special  apparatus  for  their  determination.     They  are  therefore 
neglected  in  field-work.     (See  Form  E.) 

To  reduce  m  to  a  Standard  Temperature. 

MO  =  m  (i  +  (/-  t^q) 
m$  =  value  of  ///  at  standard  temperature  /0; 
/  =  temperature  at  time  of  experiments ;  and 
q  =  temperature-constant. 

Computation  of  the  Total  Intensity. 

0  =  X  sec  0 
¥  =  total  intensity ; 
X  =  horizontal  intensity ;  and 
0  =  inclination  or  dip. 

To  convert  measures  of  intensity  expressed  in  English  units 
into  their  equivalents  expressed  in  the  metric  system,  multiply  by 
0.46108,  (log  =  9.66378.) 

To  convert  measures  of  intensity  expressed  in  metric  units  into 
their  equivalents  expressed  in  English  units,  multiply  by '2.1688, 
(log  =  0.33622.) 

Magnetic  Inclination. 

The  dip-circle  having  been  mounted  on  its  tripod  or  post,  the 
horizontal  limb  must  be  leveled.  The  needle  is  charged  by 
means  of  the  magnetizing-bars,  and  then  suspended  as  follows : 
Raise  the  Y's,  and  placing  the  needle  in  them,  lower  it  gently 
upon  the  agate  supports.  Turn  the  vertical  circle  slowly  in 
azimuth  around  the  entire  circle,  and  see  whether  the  needle 
plays  freely,  and  whether  its  face  lies  in  the  plane  of  the  face  of 
the  vertical  circle  in  all  azimuths.  If  necessary,  the  agate  supports 
must  be  re-adjusted.  The  face  of  the  vertical  circle  is  then  brought 
into  the  plane  of  the  magnetic  meridian  by  the  following  method: 
Turn  the  vertical  circle  in  azimuth  until  the  needle  is  vertical. 
Record  the  reading  of  the  azimuth-circle.  Reverse  the  needle 
on  its  supports ;  make  it  again  vertical  by  a  slight  movement  in 
azimuth,  and  record  as  before.  Turn  the  vertical  circle  180°  in 
azimuth,  and  repeat  the  double  observation.  A  mean  of  the  four 
readings  is  the  reading  of  the  magnetic  prime-vertical,  from 
which  the  settings  of  the  magnetic  meridian  are  obtained  by 


FIELD    MAGNETIC    OBSERVATIONS.  305 


LXVI.  —  Field  Magnetic  Observations,  &c.  —  Continued. 

adding  and  subtracting  90°.     Set  the  vertical  circle  at  one  of 
these  readings.     (See  Form  F.) 

The  observations  for  the  determination  of  the  inclination  are 
made  as  follows  :  Record  the  reading  of  the  azimuth-circle. 
Record  the  polarity  of  the  needle,  (marked  end  north  or  south?) 
the  position  of  the  vertical-circle,  (face  east  or  west,)  a  mean  of 
the  readings  of  the  vertical  circle  at  the  ends  of  the  needle,  and 
the  temperature  Fahrenheit.  Raise  the  needle,  reverse  it  on  its 
supports,  and  repeat  the  observations.  Bring  the  needle  back  to 
its  original  position,  and  observe  as  before.  Reverse  it  again,  and 
repeat  the  observations.  Turn  the  vertical  circle  180°  in  azimuth, 
and  repeat  the  observations.  Remove  the  needle,  and  reverse  its 
polarity.  Suspend  it  again,  and  repeat  the  observations  with  the 
circle  and  needle  in  both  positions,  as  before. 

Computation. 


0  =  magnetic  inclination  ; 
n  |  =  mean  of  observed  values  of  0  j  J^6  |  reversal  of 


polarity  ; 
c  =  constant  correction  for  errors  of  axle  and  limb. 

The  constant  correction  (c)  is  determined  as  follows  :  A  complete 
set  of  experiments  must  be  made  in  the  plane  of  the  magnetic 
meridian.  The  vertical  circle  is  then  turned  in  either  direction 
about  45°  in  azimuth,  and  a  similar  series  of  experiments  is  made. 
The  vertical  circle  is  then  turned  90°  in  azimuth  in  the  opposite 
direction,  and  the  experiments  are  again  repeated. 

Computation. 

c  =  6  —  0i-t  cot2  0  =  cot2  0'  +  cot2  0" 

6'   ) 

\  =  observed  inclinations  in  planes  at  right  angles  to 

each  other;  and 

0!  =  observed  inclination  in  the  plane  of  the  magnetic 
meridian. 


20 


306 


APPENDIX. 


FORM  A. 
Determination  of  the  Zero  of  the  Magnet- Scale. 

Station,  Fort  Yukon,  Alaska. — Date,  August  12,  (p.  m.,)  1869. — Observer, 
C.  W.  R.— Recorder,  C.  W.  R. 

Scale-zero  of  declination-magnet. 


Position  of  scale. 

Reading  of  scale. 

Alternate  means. 

Zeros. 

Mean  zero. 

Erect 

4C.    OO 

Inverted  

$7  OO 

4^   £.0 

CT    2^ 

Erect  

46.  oo 

CC.  CO 

co.  7C, 

Inverted 

£.4   OO 

A  A     7  C 

40   ^  7 

50.18 

Erect 

4^    ^O 

CC   2C. 

4.0-27 

Inverted  

^6.  c,o 

FORM  B. 
Determination  of  the  Co-efficient  of  Torsion. 

Station,  Fort  Yukon,  Alaska. — Date,  August  13,  1869.— Observer,  C.  W.  R. — 
Recorder,  C.  W.  R. 

Declination-magnet  suspended. 


Circle-readings. 

Scale-readings. 

DifT.  of  arc. 

Diff.  of  scale. 

Mean  for  90°. 

3-95 

5I-25 

° 

3°-95 
12.95 

3°-5° 
79-5° 

90 
1  80 

20.75 
49.00 

23-56 

3-95 

55-oo 

90 

24.50 

Compu  ta  tion. 


0.0     = 

tt  —  23d.  56  == 
90°  —  It  = 

H_ 
F 
.   ,   H 

699  .7 

log 
log 

log 

2.84491 

5.50961 

32330°  -3 
0.00216 
1.00216 
29".  76 

7-33530 

h  F 
*('+?)- 

H_        u 

F~      90°  —  u 

*a=    29".  70 

FIELD    MAGNETIC    OBSERVATIONS. 


307 


FORM  C. 
Declination — Record  of  Observations. 

Station,  Fort  Porter,  Buffalo,  N.  Y.,  222  feet  north  of  flag-staff. — Date,  June 
14,  1872. — Observer,  A.  N.  L. 

Azimuth-reading  at  adjustment,  ver.  B 158°  09'    oo" 

ver.  A 338    09    30 

Reading  of  mark  on  flag-staff,  .ver.  B : 166    21     oo 

ver.  A 346    21     oo 

Azimuth  of  flag-staff,  west  of  north 175     35     53 

Magnet  C2  suspended;   scale-zero,  21.7. 


Time. 

Scale. 

Remarks. 

h.     m. 

6    oo  a.  m. 

21.7 

6     20 

22.3 

6    45 

22.6 

7    oo 

22.7 

Maximum. 

7    30 

22.4 

8    oo 

22.  0 

ii     30 

15.6 

12     oo  m. 

14.75 

12     30  p.  m. 

14.35 

Minimum. 

12    45 

14.65 

I      OO 

I5-°5 

1    15 

15-7 

Computation. 


Correction  to  (5'. 

Astr.  merid  ..  161°  56'  53" 
Az.  at  adjust 
ment  .    158  09  15 

e  =  —  18 

S  =  —  21. 

5 
7 

e  —  s  =  T>.2 

<*'  =  +  3  47  38 

Corr.  to  d'  =  -}-         9  °4«  5 

r—        8. 

35 

01 

» 

e      s+f>  —  3-2S 

/—  4-     . 

fr=-\-o.  08 

T  +  F)-"56 

^  =  +  3  56  42.  5 

308 


APPENDIX. 


FORM  D. 
Horizontal  Intensity — Experiments  of  Deflection. 

Station,  Fort  Yukon,  Alaska. — Date,  August  14,  1869. — Observer,  C.  W.  R. — 
Recorder,  C.  W.  R. 

Magnets  at  right  angles  to  each  other ;  long  magnet  deflecting ;  short 
magnet  suspended. 


Magnet  at  i'.O9  east  and  west. 


W. 


3 

4 

9 

10 

15 

16 

21 

22 

Mean 


E. 
W. 


E. 
W. 
E. 
W. 
for  E. 


81.0 

82.0 
81.0 
82.0 

85.5 
87.0 
84.0 
85.0 
and 


H 


71  18 
251  18 

55  33 
235  33 

71  21 
251  21 

55  35 
235  35 


7i  34 
251  34 

55  3° 

235  30 

71  36 

251  36 

55  42 
235  42 


161  18 

H5  33 
161  21 

H5  35 

161  34 

145  3° 
161  36 

145  42 


756.6 


h.  111. 
2    IO 

2  18 

3  02 

•?       OO 

4  13 

4  22 

5  22 

5  35 


161  17.5 

145  33- 7 
161  19. 7 

145  36-4 

Mean 

161  34.  i 
145  29.8 
161  36.5 
145  41.4 
Mean 


15  43-  8 
7  51-  9 

15  46.0 
7  53-0 

15  43-3 
7  51-6 


7  52-2 


16  04.  3 
8  02.  i 

16  06.  7 
8  03.3 

15  55-1 
7  57-5 


8  01.  o 


Remarks. 


Computation.  . 


jh  34111  p.  m. — Turned  instrument  on  50.18 
declination-scale.  Verniers,  E.,  63°  51';  W., 
243°5I/-  Temperature,  78°.5  F.,  (attached.) 

3h  45m  P- ni- — Deflector  away  to  show 
changes  of  declination.  Temperature,  87°  F. 
Verniers,  E.,  63°  49';  W.,  243°  49'. 

6hoomp.m. — End  of  experiments.  Tem 
perature,  82° F.;  scale,  58.50.  Verniers,  E., 

63°  54' ;  W.,  243°  54.' 


"X 


log. 


9-  H049 
o.  11228 


9-  69897 


>•  95 i 74 


FIELD    MAGNETIC    OBSERVATIONS. 


3°9 


FORM  E. 
Horizontal  Intensity — Experiments  of  Oscillation. 

Station,  Fort  Yukon,  Alaska. — Date,  August,  16,  1869. — Chronometer,  Bliss 
&  Creighton,  1609,  M.  T. — Daily  rate,  unknown. — Observer,  C.  \V.  R. — 
Recorder,  J.  J.  M. 

Long  magnet  suspended  without  load. 


, 

£ 

t/5 

o 

°    c 

^*  •*-> 

r* 

g 

g 

u-,  -2 
o  '§ 

|l 

£ 

2 

^  1 

Remarks. 

.    r^ 

.3     O 

a 

•    • 

rH       *O 

KO    5 

QJ 

VH 

.^5         f/5 

O 

H 

Q 

H    ° 

//.  ;;/.   s. 

0 

s. 

s. 

o 

i  16  05.0 

82.5 

6 

16  40.  8 

Approximate  time  of  6  oscilla 

12 

17  16.5 

tions  at  the  beginning  35s. 

18 

17  52.3 

24 

18  28.3 

Scale   reading   noted  just   be 

30 

19  04.0 

179.0 

59.67 

fore     2OOth    oscillation     to 

36 

19  39.  7 

178.9 

59.63 

detect   changes    of    declina 

42 

20    15.5 

179.0 

59.67 

tion.      Reading,  65°.  oo;    lo 

fc    48 

20   51.3 

179.0 

59.67 

cal  time,  I2h  I5m  p.  m. 

54 

21    27.  O 

178.7 

59-57 

60 

22   02.8 

178.8 

59.60 

IOO 

26  01.5 

Si.o 

238.7 

59.67 

200 

35  57-5 

81.5 

596.0 

59-60 

Time  of  10  oscillations,  59S.64;  time  of  I  oscillation,  5s-964. 

Computation. 

q 

0.00015 

T 

Logarithms. 

f  —  t 

o°.o6 

T/2 

o.  77554 

(f  —  t)q 

o.  000009 

J  +  Fl 

1.55108 

I  -(/'-/)'/ 

0.99991 

l-(f—f)q 

o.  00103 

Logarithms. 

T, 

9.  99996 

m* 
X 

mX 

8.95174 
9.  83242 

7T2K 

niX. 

1.55207 
1.38449 

up 

8.  78416 

m 

9-  83242 

X 

9-  39208 

o.  44034 

Experiments  of  deflection. 


APPENDIX. 


FORM  F. 
Inclination — Determination  of  the  Dip. 

Station,  Willet's  Point,  N.  Y.— Date,  August  5,  1872.— Observer,  C.  W.  R.— 
Recorder,  C.  W.  R.— Dip -circle  by  Wurdemann ;  Lloyd's  needle.' 

Meridian  observations. — Settings  for  magnetic  meridian. 


Face  of  circle. 

Face  of  needle. 

Readings  of  hor.  limb. 

S. 

S. 
N. 

N. 

S. 

N. 
S. 

N. 

157  27 
158  42 
337  oi 
338  12 

Magnetic  prime-vertical,  157°  50';  settings,  247°  50',  67°  50'. 


Marked 
end. 

Face     of 
circle. 

Face    of 

needle. 

Means  of  N. 
and  S.  ends. 

Means. 

Means. 

N. 

E. 

E. 
E. 
W. 

73°  04' 
73    oo 

72      12 

73°  02'  oo" 

72°  36'  30" 

W. 

72     10 

72   ii   oo 

W. 

E. 
E. 

72    3° 
72    28 

72  29  oo 

W. 
W. 

72    49 
72    49 

72  49  oo 

72  39  oo 

Mean  ... 

N 

72  37  45 

S. 

E. 

E. 
E. 
W. 

W. 

72°  43' 
72    45 
73    oo 
73     12 

72°  44'  oo" 
73  06  oo 

72°  55'  oo" 

E. 
E. 
W. 

72    49 
72    47 
73     :4 

72  48  oo 

73  oi  oo 

W. 

73     J4 

73   H  oo 

Mean  .  .  . 

.['] 

72  58  oo 

Resulting  inclination       a~r  P 

72  47  52 

Computation. 
=  72°  47'  52";  c  =  +  18";   6  =  72°  48'. 


YC    (35S2 


